JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 9, Number 2, April 1996
A COMPACTIFICATION OVER Mg OF THE UNIVERSAL MODULI SPACE OF SLOPE-SEMISTABLE VECTOR BUNDLES
RAHUL PANDHARIPANDE
Contents 0. Introduction 425 1. The quotient construction 428 2. The fiberwise G.I.T. problem 433 3. Cohomology bounds 435 4. Slope-unstable, torsion free sheaves 437 5. Special, torsion bounded sheaves 440 6. Slope-semistable, torsion free sheaves 443 7. Two results in geometric invariant theory 449 8. The construction of Ug(e, r) 454 9. Basic properties of Ug(e, r) 458 10. The isomorphism between Ug(e, 1) and Pg,e 464 References 470
0. Introduction 0.1. Compactifications of moduli problems. Initial statements of moduli prob- lems in algebraic geometry often do not yield compact moduli spaces. For example, the moduli space Mg of nonsingular, genus g 2 curves is open. Compact moduli spaces are desired for several reasons. Degeneration≥ arguments in moduli require compact spaces. Also, there are more techniques available to study the global ge- ometry of compact spaces. It is therefore valuable to find natural compactifications of open moduli problems. In the case of Mg, there is a remarkable compactification due to P. Deligne and D. Mumford. A connected, reduced, nodal curve C of arith- metic genus g 2isDeligne-Mumford stable if each nonsingular rational component ≥ contains at least three nodes of C. Mg, the moduli space of Deligne-Mumford stable genus g curves, is compact and includes Mg as a dense open set. There is a natural notion of stability for a vector bundle E on a nonsingular curve C.Lettheslopeµbe defined as follows: µ(E)=degree(E)/rank(E). E is slope-stable (slope-semistable) if (1) µ(F ) < ( ) µ(E) ≤ Received by the editors April 18, 1994 and, in revised form, December 8, 1994. 1991 Mathematics Subject Classification. Primary 14D20, 14J10. Supported by an NSF Graduate Fellowship.
c 1996 American Mathematical Society
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for every proper subbundle F of E. When the degree and rank are not coprime, the moduli space of slope-stable bundles is open. UC(e, r), the moduli space of slope- semistable bundles of degree e and rank r,iscompact.AnopensetofUC(e, r) corresponds bijectively to isomorphism classes of stable bundles. In general, points of UC(e, r) correspond to equivalence classes (see section (1.1) ) of semistable bun- dles. The moduli problem of pairs (C, E), where E is a slope-semistable vector bundle on a nonsingular curve C, cannot be compact. No allowance is made for curves that degenerate to nodal curves. A natural compactification of this moduli problem of pairs is presented here.
0.2. Compactification of the moduli problem of pairs. let C be a fixed algebraically closed field. As before, let Mg be the moduli space of nonsingular, complete, irreducible, genus g 2 curves over the field C. For each [C] Mg, ≥ ∈ there is a natural projective variety, UC(e, r), parametrizing degree e,rankr,slope- semistable vector bundles (up to equivalence) on C.Forg 2, let Ug(e, r)betheset ≥ of equivalence classes of pairs (C, E), where [C] Mg and E is a slope-semistable vector bundle on C of the specified degree and rank.∈ A good compactification, K, of the moduli set of pairs Ug(e, r) should satisfy at least the following conditions: (i) K is a projective variety that functorially parametrizes a class of geometric objects. (ii) Ug(e, r) functorially corresponds to an open dense subset of K. (iii) There exists a morphism η : K Mg such that the natural diagram com- mutes: →
Ug(e, r) K −−−−→ η M g M g y −−−−→ y (iv) For each [C] Mg, there exists a functorial isomorphism ∈ 1 η− ([C]) ∼= UC (e, r)/Aut(C).
The main result of this paper is the construction of a projective variety Ug(e, r) that parametrizes equivalence classes of slope-semistable, torsion free sheaves on Deligne-Mumford stable, genus g curves and satisfies conditions (i-iv) above. The definition of slope-semistability of torsion free sheaves (due to C. Seshadri) is given in section (1.1).
0.3. The method of construction. An often successful approach to moduli con- structions in algebraic geometry involves two steps. In the first step, extra data is added to rigidify the moduli problem. With the additional data, the new moduli problem is solved by a Hilbert or Quot scheme. In the second step, the extra data is removed by a group quotient. Geometric Invariant Theory is used to study the quotient problem in the category of algebraic schemes. In good cases, the final quotient is the desired moduli space. In order to rigidify the moduli problem of pairs, the following data is added to (C, E): (i) An isomorphism CN+1 ∼ H0(C, ω10), → C (ii) An isomorphism Cn ∼ H0(C, E). → License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use A COMPACTIFICATION OVER Mg OF THE UNIVERSAL MODULI SPACE 427
Note ωC is the canonical bundle of C. The numerical invariants of the moduli problem of pairs are the genus g, degree e, and rank r. The rigidified problem should have no more numerical invariants. Hence, N and n must be determined by g, e,andr. Certainly, N = 10(2g 2) g by Riemann-Roch. It is assumed H1(E)=0andEis generated by global− sections.− In the end, it is checked these assumptions are consequences of the stability condition for sufficiently high degree bundles. We see n = χ(E)=e+r(1 g). − N N The isomorphism of (i) canonically embeds C in P = PC . The isomorphism of (ii) exhibits E as a canonical quotient
n C C E 0. ⊗O → → The basic parameter spaces in algebraic geometry are Hilbert and Quot schemes. Subschemes of a fixed scheme X are parametrized by Hilbert schemes Hilb(X). Quotients of a fixed sheaf F on X are parametrized by Quot schemes Quot(X, F). The rigidified curve C can be parametrized by a Hilbert scheme H of curves in PN , and the rigidified bundle E can be parametrized by a Quot scheme n Quot(C, C C) of quotients on C. In fact, Quot schemes can be defined in ⊗O a relative context. Let UH be the universal curve over the Hilbert scheme H.The n N family of Quot schemes, Quot(C, C C), defined as C, P varies in H is simply the relative Quot scheme of the⊗O universal curve over→ the Hilbert scheme: n Quot(UH H, C UH). This relative Quot scheme is the parameter space of the rigidified→ pairs (up⊗O to scalars). The Quot scheme set up is discussed in detail in section (1.3). N+1 n The actions of GLN+1(C)onC and GLn(C)onC yieldanactionof GLN+1 GLn on the rigidified data. There is an induced product action on the relative× Quot scheme. It is easily seen the scalar elements of the groups act trivially on the Quot scheme. Ug(e, r) is constructed via the quotient:
n (2) Quot(UH H, C U )/SLN+1 SLn. → ⊗O H × There is a projection of the rigidified problem of pairs (C, E) with isomorphisms (i) and (ii) to a rigidified moduli problem of curves {C with isomorphism (i) . } { } The projection is GLN+1-equivariant with respect to the natural GLN+1-action on the rigidified data of curves. The Hilbert scheme H is a parameter space of the rigidified problem of curves (up to scalars). By results of Gieseker ([Gi]) reviewed in section (1.2), the quotient H/SLN+1 is Mg. A natural morphism Ug(e, r) Mg is therefore obtained. Gieseker’s results require the choice in isomorphism (i)→ of at least the 10-canonical series. The technical heart of the paper is the study of the Geometric Invariant Theory problem (2). The method is divide and conquer. The action of SLn alone is first studied. The SLn-action is called the fiberwise G.I.T. problem. It is solved in sections (2) - (6). The action of SLN+1 alone is then considered. There are two pieces in the study of the SLN+1 action. First, Gieseker’s results in [Gi] are used in an essential way. Second, the abstract G.I.T. problem of SLN+1 acting on P(Z) × P(W )whereZ,Ware representations of SLN+1 is studied. If the linearization is taken to be P(Z)(k) P(W)(1) where k>>1, there are elementary set theoretic relationshipsO between⊗O the stable and unstable loci of P(Z)andP(Z) P(W). These relationships are determined in section (7). Roughly speaking, the abstract× lemmas are used to import the invariants Gieseker has determined in [Gi] to the problem at
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hand. In section (8), the solution of fiberwise problem is combined with the study of the SLN+1-action to solve the product G.I.T. problem (2). 0.4. Relationship with past results. In [G-M], D. Gieseker and I. Morrison propose a different approach to a compactification over Mg of the universal moduli space of slope-semistable bundles. The moduli problem of pairs is rigidified by adding only the data of an isomorphism Cn H0(C, E). By further assumptions on E, an embedding into a Grassmannian is→ obtained 0 C, G(rank(E),H (C, E)∗). → The rigidified data is thus parametrized by a Hilbert scheme of a Grassmannian. The GLn-quotient problem is studied to obtain a moduli space of pairs. Recent progress along this alternate path has been made by D. Abramovich, L. Caporaso, and M. Teixidor ([A], [Ca], [T]). A compactification, Pg,e,ofthe universal Picard variety is constructed in [Ca]. There is a natural isomorphism
ν : Pg,e Ug(e, 1). → This isomorphism is established in section (10). In the rank 2 case, the approach of [G-M] yields a compactification not equivalent to Ug(e, 2) ([A]). The higher rank constructions of [G-M] have not been completed. They are certainly expected to differ from Ug(e, r). 0.5. Acknowledgements. The results presented here constitute the author’s 1994 Harvard doctoral thesis. It is a pleasure to thank D. Abramovich and J. Harris for introducing the author to the higher rank compactification problem. Conversations with S. Mochizuki on issues both theoretical and technical have been of enormous aid. The author has also benefited from discussions with L. Caporaso, I. Morrison, H. Shahrouz, and M. Thaddeus.
1. The quotient construction 1.1. Definitions. Let C be a genus g 2, Deligne-Mumford stable curve. A coherent sheaf E on C is torsion free if ≥
x C, depth x (Ex)=1, ∀ ∈ O or equivalently, if there does not exist a subsheaf 0 F E → → such that dim(Supp(F)) = 0. Let q
C = Ci, 1 [ where the curves Ci are the irreducible components of C.Letωibe the degree of the restriction of the canonical bundle ωC to Ci.Letribe the the rank of E at the generic point of Ci.Themultirank of E is the q-tuple (r1,... ,rq). E is of uniform rank r if ri = r for each Ci.IfEis of uniform rank r, define the degree of E by e = χ(E) r(1 g). − − A torsion free sheaf E is defined to be slope-stable (slope-semistable) if for each nonzero, proper subsheaf 0 F E → → License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use A COMPACTIFICATION OVER Mg OF THE UNIVERSAL MODULI SPACE 429
with multirank (s1,... ,sq), the following inequality holds: χ(F ) χ(E) (3) q < q , 1 siωi 1 riωi respectively, P P χ(F ) χ(E) . q s ω ≤ q r ω 1 i i 1 i i The above is Seshadri’s definition of slope-(semi)stability in the case of canonical P P polarization. In case E is a vector bundle on a nonsingular curve C,theslope- (semi)stability condition (1) of section (0.1) and condition (3) above coincide. A slope-semistable sheaf has a Jordan-Holder filtration with slope-stable factors. Two slope-semistable sheaves are equivalent if they possess the same set of Jordan-Holder factors. Two equivalence classes are said to be aut-equivalent if they differ by an automorphism of the underlying curve C. For g 2 and each pair of integers (e, r 1), a projective variety Ug(e, r)and a morphism≥ ≥ η : Ug(e, r) Mg → satisfying the following properties are constructed in Theorem (8.2.1). There is a functorial, bijective correspondence between the points of Ug(e, r) and aut-equiva- lence classes of slope-semistable, torsion free sheaves of uniform rank r and degree e on Deligne-Mumford stable curves of genus g. The image of an aut-equivalence class under η is the moduli point of the underlying curve. Ug(e, r)andηwill be constructed via Geometric Invariant Theory. The G.I.T. problem is described in sections (1.2-1.7). The solution is developed in sections (2- 8) of the paper. Basic properties of Ug(e, r) are studied in section (9). In particular, the equivalence of Ug(e, 1) and Pg,e is established in section (10.3). 1.2. Gieseker’s construction. We review Gieseker’s beautiful construction of Mg.Fixagenusg 2. Define: ≥ d = 10(2g 2), − N = d g. − N Consider the Hilbert scheme Hg,d,N of genus g, degree d,curvesinPC.Let
Hg Hg,d,N ⊂ denote the locus of nondegenerate, 10-canonical, Deligne-Mumford stable curves of genus g. Hg is naturally a closed subscheme of the open locus of nondegenerate, reduced, nodal curves. In fact, Hg is a nonsingular, irreducible, quasi-projective N variety ([Gi]). The symmetries of P induce a natural SLN+1(C)-action on Hg. D. Gieseker has studied the quotient Hg/SLN+1 via geometric invariant theory. It is shown in [Gi] that, for suitable linearizations, Hg/SLN+1 exists as a G.I.T. quotient and is isomorphic to Mg.
1.3. Relative quot schemes. Let UH be the universal curve over Hg.Wehave a closed immersion N UH , Hg P → × and two projections: µ : UH Hg, → N ν : UH P . → License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 430 RAHUL PANDHARIPANDE
Let U be the structure sheaf of UH . The Grothendieck relative Quot scheme is centralO to our construction. We will be interested in relative Quot schemes of the form n (4) Quot(µ : UH Hg, C U,ν∗( N(1)),f), → ⊗O OP where f is a Hilbert polynomial with respect to the µ-relatively very ample line bundle ν∗( PN (1)). We denote the Quot scheme in (4) by Qg(µ, n, f). We recallO the basic properties of the Quot scheme. There is a canonical projective
morphism π : Qg(µ, n, f) Hg. The fibered product Qg(µ, n, f) Hg UH is equipped with two projections: → ×
θ : Qg(µ, n, f) H UH Qg(µ, n, f), × g → φ : Qg(µ, n, f) H UH UH × g → and a universal θ-flat quotient n n (5) C Q U φ∗(C U) 0. ⊗O × ' ⊗O →E→ Let ξ be a (closed) point of Qg(µ, n, f). The point π(ξ) Hg corresponds to ∈ the 10-canonical, Deligne-Mumford stable curve Uπ(ξ). Restriction of the universal quotient sequence (5) to Uπ(ξ) yields a quotient n C U ξ 0 ⊗O π(ξ) →E → with Hilbert polynomial f(t)=χ( ξ U (t)). E ⊗O π(ξ) The above is a functorial bijective correspondence between points ξ Qg(µ, n, f) n ∈ and quotients of C U ,π(ξ) Hg, with Hilbert polynomial f. ⊗O π(ξ) ∈ 1.4. Group actions. Denote the natural actions of SLN+1 on Hg and UH by:
aU UH SLN+1 U H × −−−−→ µ id µ × aH Hg SLN+1 Hg × y −−−−→ y Also define: µ inv aH µ : UH SLN+1 × H g SLN+1 H g , × −→ × −→ π id aH π : Q g ( µ, n, f) SLN+1 × H g SLN+1 H g . × −→ × −→ There is a natural isomorphism between the two fibered products
Qg(µ, n, f) SLN+1 H UH Qg(µ, n, f) H UH SLN+1 , × × g ' × g × where the projection maps to Hg are (π, µ)and(π, µ) in the first and second prod- ucts respectively. The inversion in the definition of µ is required for the isomorphism of the fibered products. There is a natural commutative diagram
aU UH SLN+1 U H × −−−−→ µ µ
id Hg Hg y −−−−→ y where id inv aU aU : UH SLN+1 × U H SLN+1 UH . × −→ × → License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use A COMPACTIFICATION OVER Mg OF THE UNIVERSAL MODULI SPACE 431
We therefore obtain a natural map of schemes over C:
% :(Qg(µ, n, f) SLN+1) H UH Qg(µ, n, f) H UH. × × g → × g By the functorial properties of Qg(µ, n, f), the %-pull-back of the universal quotient sequence (5) on Qg(µ, n, f) H UH yields a natural group action: × g Qg(µ, n, f) SLN+1 Qg(µ, n, f). × → Hence, the natural SLN+1-action on Hg lifts naturally to Qg(µ, n, f). There is a natural SLn(C)-action on Qg(µ, n, f) induced by the SLn(C)-action on the tensor n product C U. In fact, the SLN+1-action and the SLn-action commute on ⊗O Qg(µ, n, f). The commutation is most easily seen in the explicit linearized projec- tive embedding developed below in section (1.6). Hence, there exists a well-defined SLN+1 SLn-action. For suitable choices of n, f, and linearization, a component × of the quotient Qg(µ, n, f)/(SLN+1 SLn) will be Ug(e, r). × 1.5. Relative embeddings. Following [Gr], a family of relative projective em- beddings of Qg(µ, n, f)overHg is constructed. Since the inclusion N Qg(µ, n, f) H Ug , Qg(µ, n, f) P × g → × is a closed immersion, the universal quotient can be extended by zero to Qg(µ, n, f) PN .Let E × N θP:Qg(µ, n, f) P Qg(µ, n, f) × → be the projection. The universal quotient sequence (5) induces the following se- N quence on Qg(µ, n, f) P : × n 0 C Q PN 0. →K→ ⊗O × →E→ Since and Q PN are θP-flat, is θP-flat. By the semicontinuity theorems for E O × K θP-flat, coherent sheaves, there exists an integer tα such that for each t>tα and each ξ Qg(µ, n, f): ∈ 1 N (6) h(P , ξ N(t)) = 0, K ⊗OP 0 N (7) h (P , ξ N(t)) = f(t), E ⊗OP 1 N (8) h (P , ξ N(t)) = 0, E ⊗OP n 0 N 0 N (9) C H (P , N (t)) H (P , ξ N(t)) 0. ⊗ OP → E ⊗OP → The surjection of (9) follows from (6) and the long exact cohomology sequence. For each t>tα, there is a well defined algebraic morphism (on points) n 0 N it : Qg(µ, n, f) G(f(t), (C H (P , N (t)))∗) → ⊗ OP defined by sending ξ Qg(µ, n, f) to the subspace ∈ 0 N n 0 N H (P , ξ N(t))∗ (C H (P , N (t)))∗. E ⊗OP ⊂ ⊗ OP By the theorems of Cohomology and Base Change, it follows from conditions (6-8) there exists a surjection n 0 N n C H (P , PN (t)) Q θP (C Q PN(t)) θP ( PN(t)) 0, ⊗ O ⊗O ' ∗ ⊗O × → ∗ E⊗O → where θP ( PN(t)) is a locally free, rank f(t) quotient. The universal property ∗ E⊗O of the Grassmannian defines it as a morphism of schemes. It is known that there
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exists an integer tβ such that for all t>tβ, the morphism π it is a closed embedding: × n 0 N π it : Qg(µ, n, f) Hg G(f(t), (C H (P , N (t)))∗). × → × ⊗ OP The morphisms π it,t>tβ(g, n, f), form a countable family of relative projective × embeddings of the Quot scheme Qg(µ, n, f)overHg.
1.6. Gieseker’s linearization. Since the Hilbert scheme Hd,g,N is the Quot scheme N Quot(P Spec(C), N , N (1),h(s)=ds g +1), → OP OP − there are closed embeddings for s>sα: 0 N i0 :Hd,g,N G(h(s),H (P , N(s))∗). s → OP By results of Gieseker, an integer s(g) can be chosen so that the SLN+1-linearized G.I.T. problem determined by is0 has two properties: (i) Hg is contained in the stable locus. (ii) Hg is closed in the semistable locus. In order to make use of (i) and (ii) above, we will only consider immersions of the
type is0 . For each large t, there exists an immersion: 0 N n 0 N is,t : Qg(µ, n, f) G(h(s),H (P , N (s))∗) G(f(t), (C H (P , N (t)))∗). → OP × ⊗ OP By the Pl¨ucker embeddings, we obtain h(s) f(t) 0 N n 0 N js,t : Qg(µ, n, f) P( H (P , N (s))∗) P( (C H (P , N (t)))∗). → OP × ⊗ OP The fact that the SLN^+1 and SLn-actions commute^ on Qg(µ, n, f) now follows n 0 N from the observation that these actions commute on C H (P , N (t))∗. ⊗ OP 1.7. The G.I.T. problem. Let C be a Deligne-Mumford stable curve of genus g 2. For any coherent sheaf F on C, it is not hard to see: ≥ q t (10) χ(F ωC )=χ(F)+( siωi) t, ⊗ 1 · X where (s1,... ,sq) is the multirank of F ([Se]). Equation (10) and the slope in- equalities of section (1.1) yield a natural correspondence (C, E) (C, E ωt ) → ⊗ C between slope-semistable, uniform rank r, torsion free sheaves of degrees e and e + rt(2g 2). Therefore, it suffices to construct Ug(e, r)fore>>0. The strategy− for studying the G.I.T. quotient
Qg(µ, n, f)/(SLN+1 SLn) × is as follows. First a rank r 1 is chosen. Then the degree e>e(g,r)ischosen very large. The Hilbert polynomial≥ is determined by:
(11) fe,r(t)=e+r(1 g)+r10(2g 2)t. − − For [C ] Hg , fe,r(t) is the Hilbert polynomial of degree e, uniform rank r, torsion ∈ free sheaves on C with respect to N (1). The integer n is fixed by the Euler OP License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use A COMPACTIFICATION OVER Mg OF THE UNIVERSAL MODULI SPACE 433
characteristic, n = fe,r(0). As remarked in section (0.3) of the introduction, n will equal h0(C, E) for semistable pairs. Let
tˆ(g,r,e)=tβ(g,n = fe,r(0),fe,r)
be the constant defined in section (1.5) for Qg(µ, fe,r(0),fe,r). A very large t> tˆ(g,r,e) is then chosen. Selecting e and t are the essential choices that make the G.I.T. problem well-behaved. Finally, to determine a linearization of the SLN+1 × SLn-action on the image of js,t, weights must be chosen on the two projective spaces. These are chosen so that almost all the weight is on the first, h(s) 0 N (12) P( H (P , N (s))∗). OP Since SLn acts only on the second^ factor of the product, the weighting is irrelevant to the G.I.T. problem for the SLn-action alone. The SLn action is studied in sections (2)-(6). Since SLN+1 acts on both factors, the weighting is very relevant to the SLN+1- G.I.T. problem. General results of section (7) show that in the case of extreme weighting, information on the stable and unstable loci of the SLN+1-action on first factor can be transferred to the SLN+1-action on the product of the factors. Gieseker’s study of the SLN+1-action on the first factor (12) can therefore be used. In section (8), knowledge of the SLn and SLN+1 G.I.T problems is combined to solve the SLN+1 SLn G.I.T. problem on Qg(µ, n, f). × 2. The fiberwise G.I.T. problem
2.1. The fiberwise result. The fiber of π : Qg(µ, n, f) Hg over a point [C] → ∈ Hg is the Quot scheme n 10 Qg(C, n, f)=Quot(C Spec(C), C C,ω ,f). → ⊗O C For large t, the morphism it embeds Qg(C, n, f)in n t 0 10 G(f(t),(C Sym (H (C, ω )))∗). ⊗ C The embedding it yields an SLn-linearized G.I.T. problem on Qg(C, n, f). Before examining the global G.I.T. problem for the construction of Ug(e, r), we will study this fiberwise G.I.T. problem. The main result is: Theorem 2.1.1. Let g 2, r>0be integers. There exist bounds e(g,r) >r(g 1) and t(g,r,e) > tˆ(g,r,e)≥such that for each pair e>e(g,r), t>t(g,r,e) and− any [C] Hg, the following holds: ∈ Apoint ξ Qg(C, n = fe,r(0),fe,r) corresponding to a quotient ∈ n C C E 0 ⊗O → → is G.I.T. stable (semistable) for the SLn-linearization determined by it if and only if E is a slope-stable (slope-semistable), torsion free sheaf on C and n 0 0 ψ : C H (C, C ) H (C, E) ⊗ O → is an isomorphism. P. Newstead has informed the author that a generalization of this fiberwise G.I.T. problem has been solved recently by C. Simpson in [Si]. A slight twist in our approach is the uniformity of bound needed for each [C] Hg. The proof will be developed in many steps. ∈
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2.2. The Numerical Criterion. Stability for a point ξ in a linearized G.I.T. problem can be checked by examining certain limits of ξ along 1-parameter sub- groups. This remarkable fact leads to the Numerical Criterion. The most general form of the Numerical Criterion is presented in section (7.2). A more precise version for the fiberwise G.I.T. problem is stated here. Fix a vector space Z with the trivial SLn-action. In the applications below, t 0 10 Z =∼ Sym (H (C, ωC )). n Consider the linearized SLn-action on G(k, (C Z)∗) obtained from the standard n n ⊗ representation of SLn on C .Letξ G(k, (C Z)∗). The element ξ corresponds to a k-dimensional quotient ∈ ⊗ n ρξ : C Z Kξ. ⊗ → n Let v =(v1,... ,vn)beabasisofC with integer weights w =(w(v1),... ,w(vn)). For combinatorial convenience, the additional condition that the weights sum to zero is avoided here. The representation weights of the corresponding 1-parameter
subgroup of SLn are given by rescaling: ei = w(vi) i w(vi)/n.Anelement n − a C Zis said to be v-pure if it lies in a subspace of the form vi Z.The ∈ ⊗ P ⊗ weight, w(a), of such an element is defined to be w(vi). The Numerical Criterion yields: (1) ξ is unstable if and only if there exists a basis v of Cn and weights w with the following property. For any k-tuple of v-pure elements (a1,... ,ak)such that (ρξ(a1),... ,ρξ(ak)) is a basis of Kξ, the inequality
n w(v ) k w(a ) i < j n k i=1 j=1 X X is satisfied. (2) ξ is stable (semistable) if and only if for every basis v of Cn and any noncon- stant weights w the following holds. There exist v-pure elements (a1,... ,ak) such that (ρξ(a1),... ,ρξ(ak)) is a basis of Kξ and
n w(v ) k w(a ) i > ( ) j . n ≥ k i=1 j=1 X X See, for example, [N] or [M-F].
2.3. Step I. The instability arguments will use the following lemma.
Lemma 2.3.1. Let g 2, r>0,e>r(g 1) be integers, [C] Hg.Suppose ≥ − ∈ ξ Qg(C, n = fe,r(0),fe,r) corresponds to a quotient ∈ n C C E 0. ⊗O → → n 0 0 Let U C be a subspace. Let ψ(U H (C, C )) = W H (C, E).LetGbe the subsheaf⊂ of E generated by W . For any⊗ t>tˆ(Og,e,r) the⊂ following holds: if dim(U) h0(C, G ω10t) (13) > ⊗ C , n fe,r(t)
then ξ is G.I.T. unstable for the SLn-linearization determined by it.
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Proof. Let u = dim(U). Inequality (13) implies 0
3. Cohomology bounds
3.1. The bounds. In order to further investigate the fiberwise SLn-action, we need to control the first cohomology in various ways. Lemma 3.1.1. Let g 2, R>0be integers. There exists an integer b(g,R) with the following property.≥ If E is a coherent sheaf on a Deligne-Mumford stable, genus g curve C such that: (i) E is generated by global sections. (ii) E has generic rank less than R on each irreducible component of C. Then h1(C, E) Proof. Since ωC is ample and of degree 2g 2, there is a bound q(g)=2g 2on the number of components of C.SinceEis− generated by global sections and− has bounded rank, there exists an exact sequence: qR C E τ 0, 1 O → → → M where Supp(τ) has at most dimension zero. Hence 1 1 h (C, E) qR h (C, C ). ≤ · O 1 Since h (C, C )=g, b(g,R)=(2g 2)Rg + 1 will have the required property. O − License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 436 RAHUL PANDHARIPANDE Lemma 3.1.2. Let g 2, R>0be integers. Let E be a coherent sheaf on a Deligne-Mumford stable,≥ genus g curve C satisfying (i) and (ii) of Lemma (3.1.1). Suppose F is a subsheaf of E generated by global sections. Then χ(F ) < χ(E) + b(g,R). | | | | Proof. By Lemma (3.1.1), h1(C, F ) f1,... ,fm { } that contains the Hilbert polynomial of every allowed sheaf F . The morphism (14) yields a natural map ψ : Ck H0(C, F ). Let → im(ψ)=V H0(C, F ). ⊂ We note that (14) can be factored: k C C V C F 0. ⊗O → ⊗O → → Since (ii) is surjective if and only if the analogous map in which Ck is replaced by V is surjective, we can assume k h0(C, F ) h0(C, E) < χ + b(g,R). ≤ ≤ | | Suppose F is a coherent sheaf on a Deligne-Mumford stable, genus g curve satisfying: k (a) F is generated by k< χ +b(g,R) global sections: C C F 0. | | 10 ⊗O → → (b) F has Hilbert polynomial f (with respect to ωC ). Then there exists an integer t(g,k,f) such that for all t>t(g,k,f): 1 10t (i) h (C, F ωC )=0. (ii) Ck Sym⊗ t(H0(C, ω10)) H0(C, F ω10t) 0. ⊗ C → ⊗ C → License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use A COMPACTIFICATION OVER Mg OF THE UNIVERSAL MODULI SPACE 437 The existence of t(g,k,f) follows from statements (8) and (9) of section (1.5) applied to the Quot scheme Qg(µ, k, f). Now let p(g,R,χ)=max t(g,k,fj) 1 k χ +b(g,R), 1 j m . { | ≤ ≤| | ≤ ≤ } It follows easily that p(g,R,χ) has the required property. 3.2. Step II. We apply these cohomology bounds along with the Numerical Cri- terion in another simple case. First, two definitions: Define R(g,r)=r(2g 2) + 1. If E is a coherent sheaf on C with multirank (ri) − 10 and Hilbert polynomial fe,r with respect to ωC , then by (10) of section (1.7), riωi = r(2g 2). − Therefore, ri 0 τE E E0 0, → → → → where τE is the torsion subsheaf of E and E0 is torsion free. Proposition 3.2.1. Let g 2, r>0,e>r(g 1) be integers. There exists a ≥ − bound t0(g,r,e) > tˆ(g,r,e) such that for each t>t0(g,r,e),and[C] Hg the following holds: ∈ If ξ Qg(C, n = fe,r(0),fe,r) corresponds to a quotient ∈ n (15) C C E 0, ⊗O → → n 0 0 where ψ C H (C, C ) H (C, τE ) =0,thenξis G.I.T. unstable for the ⊗ O ∩ 6 SLn-linearization determined by it. Proof. Let U Cn be a 1 dimensional subspace such that ⊂ 0 0 ψ(U H (C, C )) = W H (C, τE ). ⊗ O ⊂ Let G be the subsheaf of E generated by W . For all t, 0 10t 0 10t 0 h (C, G ω ) h (C, τE ω )=h (C, τE ). ⊗ C ≤ ⊗ C By Lemma (3.1.1), 0 0 1 h (C, τE ) h (C, E)=χ(E)+h (C, E) 4. Slope-unstable, torsion free sheaves 4.1. Step III. Propositions (2.3.1) and (3.2.1) conclude G.I.T. instability from certain undesirable properties of points in Qg(C, n, fe,r). In this section, G.I.T. instability is concluded from slope-instability in the case where ψ is an isomorphism and E is torsion free. The case where ψ is not an isomorphism (and E is arbitrary) is analyzed in section (5) where G.I.T. instability is established. The above results (for suitable choices of constants and linearizations) show only points of Qg(C, n, fe,r) where ψ is an isomorphism and E is a slope-semistable torsion free sheaf may be G.I.T. semistable. The G.I.T. (semi)stability results are established in section (6). License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 438 RAHUL PANDHARIPANDE Proposition 4.1.1. Let g 2, r>0be integers. There exist bounds e1(g,r) > ≥ r(g 1) and t1(g,r,e) > tˆ(g,r,e) such that for each pair e>e1(g,r), t>t1(g,r,e) − and any [C] Hg, the following holds: ∈ If ξ Qg(C, n = fe,r(0),fe,r) corresponds to a quotient ∈ n C C E 0, ⊗O → → n 0 0 where ψ : C H (C, C ) H (C, E) is an isomorphism and E is a slope- ⊗ O → unstable, torsion free sheaf, then ξ is G.I.T. unstable for the SLn-linearization determined by it. 4.2. Lemmas and the proof. The proof of Proposition (4.1.1) requires two Lem- mas which are used to apply Lemma (2.3.1). First a destabilizing subsheaf F of E is selected. F determines a filtration: W = H0(C, F ) H0(C, E). H0(C, E)is n 1 ⊂ identified with C by ψ.LetU=ψ−(W). If F is generated by global sections, the vanishing theorems of section (3) can be applied. Riemann-Roch then shows the conditions of Lemma (2.3.1) for U, W follow from the destabilizing property of F (Lemma (4.2.2)). In fact, the vanishing argument is valid when F is generi- cally generated by global sections. Lemma (4.2.1) guarantees that a destabilizing subsheaf F generically generated by global sections exists if E is of high degree. Lemma 4.2.1. Let g 2, r>0be integers. There exists an integer e1(g,r) > ≥ r(g 1) such that for each e>e1(g,r) and [C] Hg the following holds: − ∈ If E is a slope-unstable, torsion free sheaf on C with Hilbert polynomial fe,r (with 10 respect to ωC ), then there exists a nonzero, proper destabilizing subsheaf F of E and an exact sequence (16) 0 F F τ 0, → → → → where F is generated by global sections and Supp(τ) has at most dimension zero. Proof. Since E is slope-unstable, there exists a nonzero, proper destabilizing sub- sheaf, 0 F E. → → Let C be the union of components Ci where 1 i q,andlet(si), (ri)bethe multiranks of F , E.SinceEis torsion{ free} and F ≤is nonzero,≤ the multirank of F is not identically zero. Since the Hilbert polynomial of E is fe,r, we see (by section (3.2)) R(g,r)=r(2g 2) + 1 satisfies iri The last inequality follows from the fact 0 < siωi < siωi r(2g 2). ≤ − Since F is generated by globalX sections,X Lemma (3.1.1) yields h1(C, F ) Lemma 4.2.2. Let g 2, r>0,e>e1(g,r) be integers. There exists an integer ≥ t1(g,r,e) > tˆ(g,r,e) such that for each t>t1(g,r,e) and [C] Hg, the following holds: ∈ If ξ Qg(C, n = fe,r(0),fe,r) corresponds to a quotient ∈ n C C E 0, ⊗O → → where E is a slope-unstable, torsion free sheaf on C, then there exists a nonzero, proper subsheaf 0 F E → → such that h0(C, F ) h0(C, F ω10t) (17) > ⊗ C . n fe,r(t) Proof. Let t1(g,r,e) >p(g,R(g,r),χ = fe,r(0)) be determined by Lemma (3.1.3). Take F to be a nonzero, proper, destabilizing subsheaf of E for which there exists a sequence (18) 0 F F τ 0, → → → → where F is generated by global sections and Supp(τ) has dimension zero. Such F exist by Lemma (4.2.1). Since F is a subsheaf of E and is generated by global 1 10t sections, Lemma (3.1.3) yields for any t>t1,h(C, F ωC ) = 0. By the exact 1 10t ⊗ sequence in cohomology of (18), h (C, F ωC )=0.Let(si) be the (nontrivial) multirank of F .Wehave ⊗ 0 10t h(C, F ω )=χ(F)+( siωi)10t, ⊗ C X fe,r(t)=χ(E)+r(2g 2)10t. − License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 440 RAHUL PANDHARIPANDE We obtain 0 10t χ(F ) f e,r(t) χ(E) h (C, F ω ) · − · ⊗ C = χ(F ) r(2g 2)10t χ(E) ( siωi)10t>0. · − − · The last inequality follows from the destabilizing propertyX of F . Hence χ(F ) h0(C, F ω10t) > ⊗ C . χ(E) fe,r(t) Since h0(C, F ) χ(F )andχ(E)=n, the lemma is proven. ≥ We can now apply Lemma (2.3.1). Proof. [Of Proposition (4.1.1)] Let e1(g,r) be as in Lemma (4.2.1). For e>e1(g,r), let t1(g,r,e) be determined by Lemma (4.2.2). Suppose t>t1(g,r,e). Let F be n 1 0 the subsheaf of E determined by Lemma (4.2.2). Let U C = ψ− (H (C, F )). Since ψ is an isomorphism, dim(U)=h0(C, F ). Let G be⊂ the subsheaf generated by the global sections H0(C, F ). Certainly h0(C, F ω10t) >h0(C, G ω10t). ⊗ C ⊗ C Lemmas (4.2.2) and (2.3.1) are now sufficient to conclude the desired G.I.T. insta- bility. 5. Special, torsion bounded sheaves 5.1. Lemmas. As always, let n = fe,r(0) = χ(E). If n 0 0 ψ : C H (C, C ) H (C, E) ⊗ O → is injective but not an isomorphism, then h1(C, E) = 0. We now investigate this 6 case and conclude G.I.T. instability for the corresponding points of Qg(C, n, fe,r). 1 The strategy is the following. Since H (C, E) is dual to Hom(E,ωC), the latter must be nonzero. In Lemma (5.1.1), the kernel of a nonzero element of Hom(E,ωC) is analyzed to produce a very destabilizing subsheaf F of E. Lemma (2.3.1) is then applied as in section (4). In order to carry out the above plan, the torsion of E must be treated with care. For any coherent sheaf E on C, let 0 τE E be the torsion subsheaf. E → → is said to have torsion bounded by k if χ(τE) Lemma 5.1.1. Let g 2, r>0be integers. There exists an integer e2(g,r) > ≥ r(g 1) such that for each e>e2(g,r) and [C] Hg, the following holds: − ∈ 10 If E is a coherent sheaf on C with Hilbert polynomial fe,r (with respect to ωC ) satisfying (i) h1(C, E) =0, (ii) E has torsion6 bounded by b(g,R(g,r)), then there exists a nonzero, proper subsheaf F of E with multirank (si) not identi- callyzerosuchthat (i) F is generated by global sections, (ii) χ(F ) b(g,R(g,r)) χ(E) − > +1 . siωi r(2g 2) − License or copyright restrictions may apply to redistribution;P see https://www.ams.org/journal-terms-of-use A COMPACTIFICATION OVER Mg OF THE UNIVERSAL MODULI SPACE 441 1 Proof. Since by Serre duality H (C, E)∗ ∼= Hom(E,ωC), there exists a nonzero morphism of coherent sheaves: σ : E ωC . → We have 0 σ(E ) ω C ,whereσ(E)=0.SinceωC is torsion free, σ(E)has multirank not→ identically→ zero. Note 6 0 0 χ(σ(E)) h (C, σ(E)) h (C, wC )=g. ≤ ≤ Consider the exact sequence: 0 K E σ(E) 0. → → → → Since χ(K)=χ(E) χ(σ(E)), − χ(K) χ(E) g. ≥ − For e>r(g 1) + g, χ(K) > 0andK=0.LetF be the subsheaf generated by the global sections− of K. χ(K) > 0 implies6 F =0.Letb=b(g,R(g,r)). We have 6 χ(F ) >h0(C, F ) b = h0(C, K) b χ(K) b χ(E) b g. − − ≥ − ≥ − − For e>r(g 1) + 2b + g, χ(F ) >b. Now assume e>r(g 1) + 2b + g. By the − − bound on the torsion of E, F is not contained in τE. Let (si) be the multirank of F .SinceFis not torsion, the multirank is not identically zero. In fact, since σ(E) has multirank not identically zero, 0 < siωi An analogue of Lemma (4.2.2) is now proven. Lemma 5.1.2. Let g 2, r>0,e>e2(g,r) be integers. There exists an integer ≥ t2(g,r,e) >t0(g,r,e) such that for each t>t2(g,r,e) and [C] Hg, the following holds: ∈ If ξ Qg(C, n = fe,r(0),fe,r) corresponds to a quotient ∈ n C C E 0, ⊗O → → where E is a coherent sheaf on C satisfying n 0 0 (i) ψ : C H (C, C ) H (C, E) is injective, (ii) h1(C, E⊗) =0, O → (iii) E has torsion6 bounded by b(g,(R, g, r)), License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 442 RAHUL PANDHARIPANDE n 0 then there exists a nonzero subspace W ψ C H (C, C ) generating a non- ⊂ ⊗ O zero, proper subsheaf 0 G E such that → → dim(W ) h0(C, G ω10t) (19) > ⊗ C . n fe,r(t) Proof. Let F be the subsheaf of E determined by Lemma (5.1.1). Let W = im(ψ) H0(C, F ). ∩ Since h0(C, E) <χ(E)+b(g,R(g,r)), and ψ is injective, dim(W ) >h0(C, F ) b χ(F ) b. − ≥ − Note by condition (ii) on F in Lemma (5.1.1), dim(W ) > 0. Let t>p(g,R(g,r),χ=fe,r(0)). Since F is generated by global sections, h1(C, F ω10t)=0 ⊗ C by Lemma (3.1.3). We have 0 10t h (C, F ω )=χ(F)+( siωi)10t, ⊗ C X fe,r(t)=χ(E)+r(2g 2)10t. − We compute 0 10t (χ(F ) b) fe,r(t) χ(E) h (C, F ω ) − · − · ⊗ C =(χ(F) b) r(2g 2)10t χ(E) ( siωi)10t b χ(E) − · − − · − · >r(2g 2) ( siωi) 10t b χ(E)X. − · · − · X The last inequality follows from condition (ii) on F in Lemma (5.1.1). If also t>b χ(E)=b(g,R(g,r)) (e + r(1 g)), · · − then χ(F ) b h0(C, F ω10t) − > ⊗ C . χ(E) fe,r(t) Let G be the subsheaf of F generated by W .Sincedim(W ) >χ(F) b,n=χ(E), and − h0(C, F ω10t) h0(C, G ω10t), ⊗ C ≥ ⊗ C the proof is complete. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use A COMPACTIFICATION OVER Mg OF THE UNIVERSAL MODULI SPACE 443 5.2. Step IV. Proposition 5.2.1. Let g 2, r>0be integers. There exist bounds e2(g,r) > ≥ r(g 1) and t2(g,r,e) >t0(g,r,e) such that for each pair e>e2(g,r), t>t2(g,r,e) − and any [C] Hg, the following holds: ∈ If ξ Qg(C, n = fe,r(0),fe,r) corresponds to a quotient ∈ n C C E 0 ⊗O → → 1 where h (E,C) =0,thenξis G.I.T. unstable for the SLn-linearization determined 6 by it. Proof. Let e2(g,r) be given by Lemma (5.1.1). For e>e2(g,r), let t2(g,r,e)be given by Lemma (5.1.2). Let n 0 0 ψ : C H (C, C ) H (C, E) ⊗ O → be the map on global sections. If ψ has a nontrivial kernel, ξ is unstable by Propo- sition (2.3.1). We can assume ψ is injective. Note that im(ψ) has codimension less than b(g,R(g,r)) in H0(C, E). If 0 τ E is a torsion subsheaf such that h0(C, τ)=χ(τ) b(g,R(g,r)), then → → ≥ im(ψ) H0(τ,C) =0. ∩ 6 In this case, since t>t0(g,r,e), ξ is unstable by Proposition (3.2.1). We can assume E has torsion bounded by b. We now can apply Lemma (5.1.2). Let 1 W im(ψ) be determined by Lemma (5.1.2). Let U = ψ− (W ). Since ψ is injective,⊂ dim(U)=dim(W ). Lemmas (5.1.2) and (2.3.1) now imply the desired G.I.T. instability. 6. Slope-semistable, torsion free sheaves 6.1. Step V. Let g 2, r>0 be integers. Let ≥ e>max(e1(g,r),e2(g,r)), t>max(t0(g,r,e),t1(g,r,e),t2(g,r,e)) be determined by Propositions (2.3.1, 3.2.1, 4.1.1, 5.2.1). We now conclude the only possible semistable points in the SLn-linearized G.I.T. problem determined by n t 0 10 it : Qg(C, n = fe,r(0),fe,r) G(fe,r(t), (C Sym (H (C, w )))∗) → ⊗ C are elements ξ Qg(C, n, fe,r)thatcorrespondtoquotients ∈ n C C E 0, ⊗O → → where n 0 0 ψ : C H (C, C ) H (C, E) ⊗ O → is an isomorphism and E is a slope-semistable, torsion free sheaf. In order for ξ to be semistable, ψ must be injective by Proposition (2.3.1). Surjectivity is equivalent to h1(C, E) = 0. By Proposition (5.2.1), ψ must be surjective. Since ψ is an isomorphism, E must be torsion free by Proposition (3.2.1). Finally, by Proposition (4.1.1), E must be slope-semistable. We now establish the converse. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 444 RAHUL PANDHARIPANDE Proposition 6.1.1. Let g 2, r>0be integers. There exist bounds e3(g,r) > ≥ r(g 1) and t3(g,r,e) > tˆ(g,r,e) such that for each pair e>e3(g,r), t>t3(g,r,e) − and any [C] Hg, the following holds: ∈ If ξ Qg(C, n = fe,r(0),fe,r) corresponds to a quotient ∈ n C C E 0, ⊗O → → where n 0 0 ψ : C H (C, C ) H (C, E) ⊗ O → is an isomorphism and E is a slope-stable (slope-semistable), torsion free sheaf, then ξ is a G.I.T. stable (semistable) point for the SLn-linearization determined by it. 6.2. Lemmas. For the proof of G.I.T. (semi)stability, the fundamental step is the inequality of Lemma (6.2.2) for every subsheaf F of E generated by global sections. Note this is the reverse of the inequality required by Lemma (2.3.1). Lemma (6.2.2) follows by vanishing, Riemann-Roch, and the slope-(semi)stability of E when F is nonspecial. In case h1(C, F ) = 0, an analysis in Lemma (6.2.1) utilizing Hom(F, ω) = 0 yields the required additional6 information. The Numerical Criterion of section6 (2.2) and Lemma (6.2.2) reduce the stability question to a purely combinatorial result established in Lemma (6.2.3). Lemma 6.2.1. Let g 2, r>0be integers. Let q be an integer. There exists ≥ an integer e3(g,r,q) such that for each e>e3(g,r,q) and [C] Hg, the following holds: ∈ If E is a slope-semistable, torsion free sheaf on C with Hilbert polynomial fe,r 10 (with respect to ωC )and 0 F E → → 1 is a nonzero subsheaf with multirank (si) satisfying h (C, F ) =0,then: 6 χ(F)+q χ(E) (20) < 1. siωi r(2g 2) − − Proof. Since h1(C, F ) = 0, thereP exists a nontrivial morphism 6 σ : F ωC. → Consider the subsheaf 0 σ(F ) ωC where σ(F ) = 0. By the proof of Lemma (5.1.1), χ(σ(F )) g. Consider→ the→ exact sequence 6 ≤ 0 K F σ(F ) 0. → → → → If K =0,then χ(F)+q 0 < s0 ωi < siωi r(2g 2). i ≤ − License or copyright restrictions may apply to redistribution;X see https://www.ams.org/journal-terms-of-useX A COMPACTIFICATION OVER Mg OF THE UNIVERSAL MODULI SPACE 445 We obtain: χ(F ) g r(2g 2) χ(F ) g siωi − − − si ω i · r(2g 2) 1 ≤ siωi · s ωi − − P i0 χ(K) P P . P ≤ si0 ωi Using the slope-semistability of E with respectP to K, we conclude: χ(F ) g χ(E) r(2g 2) 1 − − − . siωi ≤ r(2g 2) · r(2g 2) − − It is now clear, for largeP e depending only on g and r and q, the inequality (20) is satisfied. Lemma 6.2.2. Let g 2, r>0.Letb=b(g,R(g,r)).Lete>e3(g,r,b) > ≥ r(g 1). There exists an integer t3(g,r,e) > tˆ(g,r,e) such that for each t>t3(g,r,e) − and [C] Hg, the following holds: ∈ If ξ Qg(C, n = fe,r(0),fe,r) corresponds to a quotient ∈ n C C E 0, ⊗O → → where E is a torsion free, slope-semistable sheaf on C and 0 F E is a nonzero, proper subsheaf generated by global sections, then → → h0(C, F ) h0(C, F ω10t) ⊗ C . n ≤ fe,r(t) If E is slope-stable, h0(C, F ) h0(C, F ω10t) < ⊗ C . n fe,r(t) Proof. Suppose t>p(g,R(g,r),χ = fe,r(0)). Let (si) be the nontrivial multirank of F .SinceFis generated by global sections, the vanishing guaranteed by Lemma (3.1.3) yields 0 10t h (C, F ω )=χ(F)+( siωi)10t. ⊗ C The Hilbert polynomial can be expressed: X fe,r(t)=χ(E)+r(2g 2)10t. − First consider the case where h1(C, F )=0.Then h0(C, F )=χ(F). We compute 0 10t χ(F ) fe,r(t) χ(E) h (C, F ω ) · − · ⊗ C = χ(F ) r(2g 2)10t χ(E) ( siωi)10t<( )0, · − − · ≤ where E is slope-stable, (slope-semistable). HenceX h0(C, F ) χ(F ) h0(C, F ω10t) = < ( ) ⊗ C . χ(E) χ(E) ≤ fe,r(t) Since n = χ(E), the nonspecial case is thus settled. Now suppose h1(C, F ) =0. Lemma (6.2.1) now applies to F . We compute 6 0 10t (χ(F )+b) fe,r(t) χ(E) h (C, F ω ) · − · ⊗ C =(χ(F)+b) r(2g 2)10t χ(E) ( siωi)10t + b χ(E) · − − · · < ( siωi) r(2g 2) 10t + b χ(EX). − · − · · License or copyright restrictions may applyX to redistribution; see https://www.ams.org/journal-terms-of-use 446 RAHUL PANDHARIPANDE For t>b χ(E)=b (e+r(1 g)), · · − χ(F )+b h0(C, F ω10t) < ⊗ C . χ(E) fe,r(t) Since h0(C, F ) <χ(F)+b, h0(C, F ) χ(F )+b < . χ(E) χ(E) The proof is complete. We require a simple combinatorial lemma. Lemma 6.2.3. Let n 2 be an integer. Let ≥ W1 W2 ... Wn,W1 Proof. Let e3(g,r)=e3(g,r,b(g,R(g,r))) be determined by Lemma (6.2.1). For e>e3(g,r), let t3(g,r,e) >p(g,R(g,r),χ = fe,r(0)) be given by Lemmas (6.2.2) and (3.1.3). We will apply the Numerical Criterion to the linearized SLn-action on n t 0 10 G(fe,r(t), (C Sym (H (C, ω )))∗). ⊗ C The element ξ corresponds to the quotient: ψt : Cn Symt(H0(C, ω10)) H0(C, E ω10t) 0. ⊗ C → ⊗ C → n Let v =(v1,... ,vn)beabasisofC . Let (w(v1),... ,w(vn)) be weights satisfying w(v1) w(v2) ... w(vn),w(v1) 0 10t Define for 1 i n, Ai = h (C, Fi ω ). ≤ ≤ ⊗ C The required fe,r(t)-tuple (a1,a2,... ,afe,r (t)) is constructed as follows. Select elements t 0 10 (a1,... ,aA1) v1 Sym (H (C, ωC )) t t ∈ ⊗ 0 10t such that (ψ (a1),... ,ψ (aA )) determines a basis of H (C, F1 ω ). Select 1 ⊗ C t 0 10 (aA +1,... ,aA ) v2 Sym (H (C, ω )) 1 2 ∈ ⊗ C t t 0 10t such that (ψ (a1),... ,ψ (aA2)) determines a basis of H (C, F2 ωC ). Continue selecting ⊗ t 0 10 (aAi+1,... ,aAi+1 ) vi+1 Sym (H (C, ωC )) t t ∈ ⊗ 0 10t such that (ψ (a1),... ,ψ (aAi+1 )) determines a basis of H (C, Fi+1 ωC ). Note t t ⊗ that if Ai = Ai+1,then(ψ(a1),... ,ψ (aAi)) already determines a basis of 0 10t t 0 10 H (C, Fi+1 ωC ) and no elements of vi+1 Sym (H (C, ωC )) are chosen. This selection is possible⊗ by the surjectivity of (21).⊗ Let α1 = A1/fe,r(t)andαi =(Ai Ai 1)/fe,r(t)for2 i n.Wehave − − ≤ ≤ f (t) n e,r w(a ) α w(v )= j . i i f (t) i=1 j=1 e,r X X Let βi =(1/n). Note n n βi = αi =1. 1 1 X X 0 Let 1 m n 1. Since ψ is an isomorphism, m h (C, Fm). Suppose Fm = E. Then≤ Lemma≤ (6.2.2)− yields ≤ 6 m A (22) < ( ) m . n ≤ fe,r(t) If Fm= E,thenAm =fe,r(t) and the inequality (22) holds trivially (m n 1). So for all 1 m n 1, we have ≤ − ≤ ≤ − m m βi < ( ) αi. 1 ≤ 1 X X Lemma (6.2.3) yields n n n fe,r (t) w(vi) w(aj ) = βiw(vi) > ( ) αiw(vi)= . n ≥ f (t) i=1 1 1 j=1 e,r X X X X By the Numerical Criterion, ξ is G.I.T. stable (semistable). 6.4. Step VI. Only one step remains in the proof of Theorem (2.1.1). It must be checked that strict slope-semistability of E implies strict G.I.T. semistability. Lemma 6.4.1. Let g 2, r>0be integers. There exists an integer e4(g,r) such ≥ that for each e>e4(g,r) and [C] Hg, the following holds: If E is any slope-semistable, torsion∈ free sheaf on a C with Hilbert polynomial 10t fe,r (with respect to ω )and0 F Eis a nonzero subsheaf with multirank C → → (si) satisfying χ(F ) χ(E) (23) = , siωi r(2g 2) − License or copyright restrictions may apply to redistribution;P see https://www.ams.org/journal-terms-of-use 448 RAHUL PANDHARIPANDE then (i) h1(C, F )=0. (ii) F is generated by global sections. Proof. Suppose F is a nonzero subsheaf of E satisfying (23). If e>e3(g,r,0), by Lemma (6.2.1), h1(C, F )=0.Nowletx Cbe a point. We have an exact sequence ∈ F 0 mxF F 0. → → → mxF → Since F is torsion free and C is nodal, it is not hard to show that F dim < 2 R(g,r). m F · x Since (F/mxF) is torsion, mxF has the same multirank as F .Also χ(mxF)>χ(F) 2R. − By (23), χ(m F )+2R χ(E) x > . siωi r(2g 2) 1 − If e>e3(g,r,2R), h (C, mxF )P = 0 by Lemma (6.2.1). In this case F is generated by global sections. We can therefore choose e4(g,r)=e3(g,r,2R). Proposition 6.4.1. Let g 2, r>0be integers. There exist bounds e4(g,r) and ≥ t4(g,r,e) such that for each pair e>e4(g,r), t>t4(g,r,e) and any [C] Hg,the following holds: ∈ If ξ Qg(C, n = fe,r(0),fe,r) corresponds to a quotient ∈ n C C E 0, ⊗O → → where n 0 0 ψ : C H (C, C ) H (C, E) ⊗ O → is an isomorphism and E is a strictly slope-semistable, torsion free sheaf, then ξ is a G.I.T. strictly semistable point for the SLn-linearization determined by it. Proof. Since ξ is G.I.T. semistable for e>e3(g,r)andt>t3(g,r,e), it suffices to find a semistabilizing 1-parameter subgroup. If e>e4(g,r), then, for any nonzero, proper semistabilizing subsheaf 0 F E,wehaveh0(C, F )=χ(F)andF is generated by global sections. It is→ now→ easy to see that the flag 0 H0(C, F ) 0 ⊂ ⊂ H (C, E)withweights 0,1 determines semistabilizing data for large t>t4(g,r,e). { } We have now shown for the bounds: e(g,r)=max ei(g,r) 1 i 4 , { | ≤ ≤ } t(g,r,e)=max ti(g,r,e) 0 i 4 { | ≤ ≤ } the claim of Theorem (2.1.1) holds. This completes the proof of Theorem (2.1.1). By Lemma (6.4.1), each slope-semistable, torsion free sheaf E on C with Hilbert SS polynomial fe,r appears in Qg(C, fe,r(0),fe,r)t for e>e(g,r), t>t(g,r,e). It is SS now clear the SLn-orbits of Qg(C, fe,r (0),fe,r)t correspond exactly to the slope- semistable, torsion free sheaves on C with Hilbert polynomial fe,r. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use A COMPACTIFICATION OVER Mg OF THE UNIVERSAL MODULI SPACE 449 6.5. Seshadri’s construction. In [Se], C. Seshadri has studied the SLn-action on Qg(C, n = fe,r(0),fe,r) via a covariant construction. For e>>0, he finds a G.I.T. (semi)stable locus that coincides exactly with the G.I.T. (semi)stable locus of Theorem (2.1.1). These results appear in Theorem 18 of chapter 1 of [Se] for nonsingular curves and Theorem 16 of chapter 6 for singular curves. The collapsing of semistable orbits is determined by the Zariski topology. Seshadri shows that (i) If Et is a flat family of slope-semistable, torsion free sheaves on C such that the Jordan-Holder factors of Et=0 are Fj , then the Jordan-Holder factors 6 { } of E0 are also Fj . (ii) If E is a slope-semistable,{ } torsion free sheaf on C with Jordan-Holder factors Fj , then there exists a flat family of slope-semistable, torsion free sheaves { } Et such that: Et=0 = E, E0 = Fj . 6 ∼ ∼ j M Statement (ii) is proven by constructing flat families over extension groups. It follows from these two results that the points of our quotient SS Qg(C, n = fe,r(0),fe,r)t /SLn for e>e(g,r),t>t(g,r,e) naturally parametrize slope-semistable, torsion free sheaves with Hilbert polynomial fe,r up to equivalence given by Jordan-Holder factors. 7. Two results in geometric invariant theory 7.1. Statements. Let V , Z,andW be finite dimensional C-vector spaces. Con- sider two rational representations of SL(V ): ζ : SL(V ) SL(Z), → ω : SL(V ) SL(W). → These representations define natural SL(V )-linearized actions on P(Z)andP(W). There is an induced SL(V )-action on the product P(Z) P(W ). Since × Pic(P(Z) P(W)) = Z Z, × ⊕ there is a 1-parameter choice of linearization. For a, b N+, let [a, b]denotethe ∈ linearization given by the line bundle P(Z)(a) P(W)(b). Subscripts will be used to indicate linearization. Let O ⊗O ρZ : P(Z) P(W ) P(Z) × → be the projection on the first factor. Proposition 7.1.1. There exists an integer kS(ζ,ω) such that for all k>kS: 1 S S ρ− (P(Z) ) (P(Z) P(W)) . Z ⊂ × [k,1] Proposition 7.1.2. There exists an integer kSS(ζ,ω) such that for all k>kSS: SS 1 SS (P(Z) P(W)) ρ− (P(Z) ). × [k,1] ⊂ Z D. Edidin has informed the author that Proposition (7.1.1) is essentially equiv- alent to Theorem 2.18 of [M-F]. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 450 RAHUL PANDHARIPANDE 7.2. q-stability. Let λ : C∗ SL(V ) be a 1-parameter subgroup. Let dim(V )= → a. It is well known there exists a basis v =(v1,... ,va)ofV such that λ takes the form ei λ(t)(vi)=t vi,tC∗. · ∈ Denote the tuple (e1,... ,ea)bye. The exponents satisfy the determinant 1 condi- a tion, ei =0.Let e =max ei . For the representation ζ : SL(V ) SL(Z), i=1 | | {| |} → there exists a basis z =(z1,... ,zb) such that ζ λ takes the form P ◦ fj ζ λ(t)(zj )=t zj,tC∗. ◦ · ∈ b Again, j=1 fj =0.The pairs v, e and z,f are said to be diagonalizing data for λ and ζ λ respectively. { } { } Let [Pz] ◦P(Z) correspond to the one dimensional subspace of Z spanned by z = 0. By∈ the Mumford-Hilbert Numerical Criterion, [z] is a stable (semistable) point6 for the ζ-induced linearization on P(Z) if and only if for every 1-parameter subgroup λ : C∗ SL(V ), the following condition holds: let z,f be diagonalizing → b { } data for ζ λ and let z = ξj zj , then there exists an index j for which ξj =0 ◦ j=1 · 6 and fj < 0(fj 0). Let q>0bearealnumber.Thepoint[≤ P z] is defined to be q-stable for the ζ- induced linearization if and only if for every 1-parameter subgroup λ : C∗ SL(V ) the following condition holds: let v, e and z,f be diagonalizing data→ for λ and b { } { } ζ λ and let z = ξj zj, then there exists an index j for which ξj =0and ◦ j=1 · 6 f < q e. For q>0, let P(Z)qS denote the q-stable locus for the ζ-induced j P linearization.− ·| | Proposition (7.1.1) will be established in two steps: Lemma 7.2.1. There exists q(ζ) > 0 such that P(Z)qS = P(Z)S. Lemma 7.2.2. For any q>0, there exists an integer kqS(q, ω) such that for all k>kqS: 1 qS S ρ− (P(Z) ) (P(Z) P(W)) . Z ⊂ × [k,1] Lemmas (7.2.1) and (7.2.2) certainly imply Proposition (7.1.1). 7.3. Proofs of Lemmas (7.2.1) and (7.2.2). Let U be a finite dimensional Q- vector space. Let L = li be a finite set of elements of U ∗.ThesetLis said to be a stable configuration if{ } 0 = u U, ili(u)<0. ∀ 6 ∈ ∃ 0 If u =(u1,u2,... ,ud)isabasisofU, define a norm u :U Q≥ by || → k u u = max γi , where u = γiui. | | {| |} 1 X Lemma 7.3.1. Suppose L = li is a stable configuration in U.Letube a basis of U. Then there exists q>0depending{ } upon L and u such that (24) 0 = u U, ili(u)< q uu. ∀ 6 ∈ ∃ −·| | Proof. Let U UR = U Q R. Suppose there exists an element 0 = u UR and ⊂ ∼ ⊗ 6 ∈ a decomposition L = L0 L00 satisfying: ∪ (i) l L0,l(u)=0. ∀ ∈ (ii) l L00,l(u)>0. ∀ ∈ License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use A COMPACTIFICATION OVER Mg OF THE UNIVERSAL MODULI SPACE 451 Since the locus z UR l L0,l(z)=0 is a rational subspace and the locus { ∈ |∀ ∈ } z UR l L00,l(z)>0 is open, there must exist an element 0 =ˆu U {satisfying∈ |∀ (i) and∈ (ii). Since L}is a stable configuration in U,suchˆudo not6 exist.∈ It follows 0 = u UR, ili(u)<0. ∀ 6 ∈ ∃ Let S be the unit box in UR: S = u UR uu =1 .Define a function g : S R− by { ∈ || | } → g(s)=min l(s) l L . { | ∈ } The function g is continuous and strictly negative. Since S is compact, g achieves a maximum value m on S for some m>0. The bound q = m/2 clearly satisfies (24). − Proof. [Of Lemma (7.2.1)] The proof consists of two simple pieces. First, a basis v of V is fixed. By applying Lemma (7.3.1), it is shown there exists a q>0 such that the stability of [z] implies the q-stability inequality for all 1-parameter subgroups of SL(V ) diagonal with respect v. Second, it is checked that this q suffices for any selection of basis. Let v =(v1,... ,va)beabasisofV.Let a U= (e1,... ,ea) ei Q, ei =0 . { | ∈ 1 } X There exist linear functions l1,... ,lb on U and a basis z =(z1,... ,zb)ofZ { } satisfying the following: if λ : C∗ SL(V ) is any 1-parameter subgroup with → diagonalizing data (v,e), then the diagonalizing data of ζ λ is (z,(l1(e),... ,lb(e))). ◦ Let L1,... ,LB be the set of distinct stable configurations in l1,... ,lb .That { } { } is, for all 1 J B, LJ l1,... ,lb and LJ is a stable configuration in U.Let ≤ ≤ ⊂{ } u=(u1,... ,ua 1)beabasisofUof the following form: − u1 =( 1,1,0,... ,0), ... , ua 1 =( 1,0,... ,0,1). − − − Note e a eu for e U. By Lemma (7.3.1), there exists qJ > 0 such that (24) | |≤ ·| | ∈ holds for each stable configuration LJ .Let 1 q= min qJ . a· { } S b Suppose [z] P(Z) .Letz= ξjzj. By the Numerical Criterion, the stability ∈ 1 of [z] implies the set lj ξj =0 is a stable configuration in U equal to some LJ .For any 1-parameter subgroup{ | 6 with}P diagonalizing data (v,e), the diagonalizing data of ζ λ is (z,(l1(e),... ,lb(e))). By (24), we see there exists an li LJ such that ◦ ∈ li(e) < qJ eu q e. − ·| | ≤− ·| | Suppose v0 is another basis of V . Then, up to scalars, there exists an element γ SL(V ) satisfying γ(v)=v0. It is now clear that ∈ (ζ(γ)(z), (l1(e),... ,lb(e))) is diagonalizing data for ζ λ where λ has diagonalizing data (v0, e). Since the ◦ set l1,... ,lb is independent of v, the above analysis is valid for any 1-parameter subgroup.{ We} have shown that [z] P(Z)qS. ∈ License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 452 RAHUL PANDHARIPANDE Lemma 7.3.2. Let ω : SL(V ) SL(W) be a rational representation. There → exists an Mω > 0 with the following property. Let λ : C∗ SL(V ) be any 1- parameter subgroup. Let (v,e) and (w, h) be diagonalizing data→ for λ and ω λ. ◦ Then h License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use A COMPACTIFICATION OVER Mg OF THE UNIVERSAL MODULI SPACE 453 Lemma 7.4.1. Consider the ζ-linearized G.I.T. problem on P(Z). There exists a finite set of a-tuples, , such that P P(Z)eUN = P(Z)UN. e [∈P Proof. We first show that P(Z)eUN is a constructible subset of P(Z). Fix a 1- parameter subgroup λ : C∗ SL(V ) with diagonalizing data (v,e). Let (z,f)be diagonalizing data for ζ λ.Let→ Hbe the projective subspace of P(Z) spanned by ◦ eUN the set zj fj > 0 . Certainly H P(Z) . Since every 1-parameter subgroup of SL(V{) with| diagonalized} exponents⊂ e is conjugate to λ,weseethemap κ:SL(V ) H P(Z) × → defined by: κ(y,[z]) = [ζ(y)(z)] is surjective onto P(Z)eUN . The unstable locus, P(Z)UN, is closed. Also, P(Z)UN is the countable union of the P(Z)eUN . Over an uncountable algebraically closed field, any algebraic variety that is countable union of constructible subsets is actually the union of finitely many of them. Therefore a finite set of a-tuples, , with the demanded property exists in the case C is uncountable. P There always exists a field extension C C0 where C0 is an uncountable alge- braically closed field. By base extension, → ζC : SLC (V C C0) SLC (Z C C0). 0 0 ⊗ → 0 ⊗ Since C is algebraically closed, it is easy to see that the C-valuedclosedpointsof eUN eUN PC0(Z CC0) are simply the points of PC(Z) . Hence, the assertion for C0 implies⊗ the assertion for C. This settles the general case. Proof. [Of Lemma (7.1.2)] Let be determined by Lemma (7.4.1) for the repre- P sentation ζ.LetMωbe determined by Lemma (7.3.2) for the representation ω.Let Nζ satisfy e ,Nζ>e. ∀ ∈P || Let kSS = Mω Nζ. Suppose k>kSS. For each element · [z] [w] P(Z)UN P(W), × ∈ × we must show that [zk w] is unstable for the Symk(ζ) ω-induced linearization on P(Symk(Z) W).⊗ Since [z] P(Z)UN,thereexistsan⊗ e such that [z] ⊗ ∈ ∈P is e-unstable for the ζ-induced linearization on P(Z). Let λ : C∗ SL(V )be a 1-parameter subgroup with diagonalized exponents e that destabilizes→ [z]. Let b (z,f)and(w, h) be diagonalizing data for ζ λ and ω λ.Letz= 1ξs zsand c ◦ ◦ · w = σt wt be the basis expansions. Since λ destabilizes [z], we see 1 · P (25) ξs =0 fs >0. P 6 ⇒ A diagonalizing basis of Symk(ζ) λ can be constructed by taking homogeneous monomials of degree k in z. Denote◦ this basis with the corresponding exponents by (z∗, f ∗). Then z∗ w is a diagonalizing basis of ⊗ (Symk(ζ) ω) λ. ⊗ ◦ We must show that every nonzero coefficient of the expansion of zk w in the basis ⊗ z∗ w corresponds to a positive exponent. Suppose the basis element z∗ wt has a ⊗ s∗ ⊗ License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 454 RAHUL PANDHARIPANDE nonzero coefficient. The element z must correspond to a homogeneous polynomial s∗∗ of degree k in those zs for which ξs = 0. Therefore, by (25), the exponent f ∗ is 6 s∗ not less than k. The exponent corresponding to z∗ wt is f ∗ + ht.Since s∗ ⊗ s∗ ht h 8. The construction of Ug(e, r) 8.1. Uniform rank. Define r Q (µ, n, fe,r) Qg(µ, n, fe,r) g ⊂ to be the subset corresponding to quotients n C C E 0, ⊗O → → r where E has uniform rank r on C. Certainly Qg(µ, n, fe,r)isSLN+1 SLn - invariant. × r r Lemma 8.1.1. Qg(µ, n, fe,r) is open and closed in Qg(µ, n, fe,r).(Qg(µ, n, fe,r) is a union of connected components.) Proof. Let κ : be a projective, flat family of Deligne-Mumford stable genus g curves over anC→B irreducible curve. Let be a κ-flat coherent sheaf of constant Hilbert 10E polynomial fe,r (with respect to ω / ). Suppose there exists a b∗ such that C B ∈B b has uniform rank r on b = C.Let i be the irreducible components of . E ∗ C ∗ {C } C Since κ : i is surjective of relative dimension 1, each i contains a component C →B C of C. Since the function r(z)=dim ( k(z)) is upper semicontinuous on i, k(z) E⊗ C there is an open set Ui i where r(z) r. Itfollowsthereexistsanopenset ⊂C ≤ U such that b U, the rank of b on each component of b is at most r. ⊂B ∀ ∈ E C If b0 such that b is not of uniform rank r, then (by semicontinuity) i so that ∃ E 0 ∃ r(z) is strictly less than r on an open W i.Forbin the nonempty intersection ⊂C U κ(W ), ranks of b are at most r on each component and strictly less than r on ∩ E at least one component. By equation (10) of section (1.7), b can not have Hilbert E polynomial fe,r.Thus b , b has uniform rank r. The Lemma is proven. ∀ ∈B E 8.2. Determination of the semistable locus. Select e>e(g,r)andt>t(g,r,e). As usual, let n = fe,r(0). Let h(s) 0 N Z = H (P , N (s))∗, OP fe,r (t^) n 0 N W = (C H (P , N (t)))∗. ⊗ OP Consider the immersion ^ h(s) fe,r (t) r 0 N n 0 N js,t : Q (µ, n, fe,r) P( H (P , N (s))∗) P( (C H (P , N (t)))∗) g → OP × ⊗ OP defined in section (1.6). Recall^ s is the linearization determined^ by Gieseker. There are three group actions to examine. In what follows, the superscripts S0,SS0 { } will denote stability and semistability with respect to the SLN+1-action. Similarly, S00,SS00 will correspond to the SLn-action, and S, SS will correspond to the { } { } SLN+1 SLn-action. × License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use A COMPACTIFICATION OVER Mg OF THE UNIVERSAL MODULI SPACE 455 The strategy for obtaining the desired SLN+1 SLn-semistable locus is as fol- × lows. Consider first the SLN+1-action. For suitable linearization, it will be shown r that Qg(µ, n, fe,r) is contained in the SLN+1-stable locus and is closed in SLN+1- semistable locus. This assertion is a consequence of Gieseker’s conditions on Hg ((i), (ii) of section (1.6) ) and the results of section (7). Next, Ug(e, r) is defined r SS as the G.I.T. quotient of Q (µ, n, fe,r) by SLN+1 SLn. Ug(e, r) is a projective g × variety. Finally, in Proposition (8.2.1), it is shown that the SLn and SLN+1 SLn r × semistable loci coincide on Qg(µ, n, fe,r). Similarly, the stable loci coincide. The results on the fiberwise G.I.T. problem now yield a geometric identification of the stable and semistable loci for the SLN+1 SLn- G.I.T. problem. × By Propositions (7.1.1) and (7.1.2), an integer k> kS0,kSS0 can be found so that: { } 1 S0 S0 (26) ρ− (P(Z) ) (P(Z) P(W )) , Z ⊂ × [k,1] SS0 1 SS0 (27) (P(Z) P(W )) ρ− (P(Z) ). × [k,1] ⊂ Z S0 By (i) of section (1.6), Hg P(Z) . Now (26) above yields ⊂ S0 (28) Hg P(W ) (P(Z) P(W )) . × ⊂ × [k,1] Therefore, r S0 (29) Q (µ, n, fe,r) (P(Z) P(W )) . g ⊂ × [k,1] SS0 By (ii) of section (1.6), Hg is closed in P(Z) . Hence Hg P(W )isclosedin 1 SS0 r × ρZ− (P(Z) ). By (27) and the projectivity of Qg(µ, n, fe,r)overHg: r SS0 Q(µ, n, fe,r)isclosedin(P(Z) P(W)) . g × [k,1] Since (P(Z) P(W ))SS (P(Z) P(W ))SS0 , × [k,1] ⊂ × [k,1] it follows that r SS SS (30) Q (µ, n, fe,r) is closed in (P(Z) P(W )) . g [k,1] × [k,1] We define r SS Ug(e, r)=Q (µ, n, fe,r) /(SLN+1 SLn). g [k,1] × By (30), Ug(e, r) is a projective variety. r SS We now identify the locus Qg(µ, n, fe,r)[k,1]. Certainly r S r S00 Q (µ, n, fe,r) Q (µ, n, fe,r) , g [k,1] ⊂ g [k,1] r SS r SS00 Q (µ, n, fe,r) Q (µ, n, fe,r) . g [k,1] ⊂ g [k,1] In fact: Proposition 8.2.1. There are two equalities: r S r S00 Qg(µ, n, fe,r)[k,1] = Qg(µ, n, fe,r)[k,1], r SS r SS00 Qg(µ, n, fe,r)[k,1] = Qg(µ, n, fe,r)[k,1]. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 456 RAHUL PANDHARIPANDE Proof. We apply the Numerical Criterion. Let k k ζ : SLN+1 SLn SLN+1 SL(Sym (Z)), × → → ω : SLN+1 SLn SL(W) × → r S00 denote the two representations. Let ξ Q (µ, n, fe,r) . Recalling the morphisms ∈ g [k,1] defined in section (1.5), js,t(ξ)=(π(ξ),it(ξ)). Let λ : C∗ SLN+1 SLn be a nontrivial 1-parameter subgroup given by → × λ1 : C∗ SLN+1, → λ2 : C∗ SLn. k→ Let mi be a diagonalizing basis for ζ λ with weights w(mi) .Let nj be { } ◦ { } { } a diagonalizing basis for the C∗ C∗ representation ω (λ1 λ2). Let w1(nj ) × ◦ × and w2(nj) denote the weights of the induced C∗ representations ω (λ1 1) and ◦ × ω (1 λ2). The weights w(nj ) of the C∗ representation ω λ are given by ◦ × { } ◦ w(nj)=w1(nj)+w2(nj). Finally let mi and nj denote the elements of the diagonalizing bases that appear with nonzero{ } coefficient{ } in the expansions of π(ξ) and it(ξ). There are three cases. (1) λ1 =1.Sinceξis a stable point for the SLn-action, there is a nj with w2(nj ) < 0. We see w(m i nj )=w(mi)+w1(nj)+w2(nj)=w2(nj)<0 ⊗ for any mi. (2) λ2 = 1. By (29), ξ is a stable point for the SLN+1-action. Hence there exists apairmi,nj so that w(mi nj)=w(mi)+w1(nj)<0. ⊗ (3) λ1 =1,λ2=1.Sinceξis a stable point for the SLn-action, there is a nj 6 6 with w2(nj ) < 0. By (28), (π(ξ) nj) is a stable point for the SLN+1-action. ⊗ Hence there exists an element mi so that w(mi)+w1(nj)<0. Therefore, w(mi nj)=w(mi)+w1(nj)+w2(nj)<0. ⊗ r S By the Numerical Criterion, ξ Q (µ, n, fe,r) . The proof for the semistable ∈ g [k,1] case is identical. r SS By Theorem (2.1.1), we see the points of Qg(µ, n, fe,r)[k,1] correspond exactly to quotients n C C E 0, ⊗O → → where E is slope-semistable, torsion free sheaf of uniform rank r on a 10-canonical, Deligne-Mumford stable, genus g curve C PN and ⊂ n 0 0 ψ : C H (C, C ) H (C, E) ⊗ O → r S is an isomorphism. Similarly for Qg(µ, n, fe,r)[k,1]. r SS We now examine orbit closures. Suppose ξ Qg(µ, n, fe,r)[k,1] lies in the r SS ∈ SLN+1 SLn-orbit closure of ξ Q (µ, n, fe,r) .Let × ∈ g [k,1] γ=(γ1,γ2): p SLN+1 SLn 4−{ }→ × License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use A COMPACTIFICATION OVER Mg OF THE UNIVERSAL MODULI SPACE 457 be a morphism of a nonsingular, pointed curve such that Limz p(γ(z) ξ)=ξ. → · It follows that Limz p(γ1(z) π(ξ)) = π(ξ). → · S Since Hg P(Z) , π(ξ) lies in the SLN+1-orbit of π(ξ). After a possible base ⊂ change, we can assume γ1 extends over p to ∈4 γ1 : p SLN+1. 4 → r Since Qg(Uπ(ξ),n,fe,r) is projective, r µ : p Q (U ,n,fe,r) 4−{ }→ g π(ξ) defined by µ(z)=γ2(z) ξextends to p.Letξˆ=Limz p(γ2(z) ξ)=µ(p).By considering the map · 4 → · r γ1 µ : p Q (µ, n, fe,r) · 4 → g defined by γ1 µ(z)=γ1(z) µ(z) · · we obtain γ1(p) ξˆ = γ1 µ(p)=Limz p(γ1(z) γ2(z) ξ) · · → · · = Limz p(γ(z) ξ)=ξ. → · We have shown the SLN+1-orbit of ξ meets the SLn-orbit closure of ξ.Ifξ,ξ correspond to a slope-semistable, torsion free quotients E, E on C, C PN , ⊂ certainly C, C must be projectively equivalent. The elements of the SLN+1-orbit of ξ that lie over C are simply the images of E under automorphisms of C.Now from section (6.5), we conclude two semistable orbits ξ and ξ are identified in the quotient Ug(e, r) if and only if π(ξ) π(ξ) [C] ≡ ≡ and the corresponding semistable, torsion free quotient sheaves E, E have Jordan- Holder factors that differ by an automorphism of C. We see: Theorem 8.2.1. Ug(e, r) parametrizes aut-equivalence classes of slope-semistable, torsion free sheaves of uniform rank r and degree e on Deligne-Mumford stable curves of genus g. Finally, since r SS Q (µ, n, fe,r) Hg Mg g [k,1] → → is an SLN+1 SLn invariant morphism, there exists a map × − η : Ug(e, r) Mg. → License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 458 RAHUL PANDHARIPANDE 9. Basic properties of Ug(e, r) 9.1. The functor. Let g(e, r) be the functor that associates to each scheme S the set of equivalence classesU of the following data: (1) A flat family of Deligne-Mumford stable genus g curves, µ : S. (2) A µ-flat coherent sheaf on such that: C→ E C 10 (i) is of constant Hilbert polynomial fe,r (with respect to ω ). E /S (ii) is a slope-semistable, torsion free sheaf of uniform rank rCon each fiber. E Two such data sets are equivalent if there exists an S isomorphism φ : 0 and C→C a line bundle L on S so that = φ∗( 0) µ∗L. E ∼ E ⊗ Theorem 9.1.1. There exists a natural transformation φU : g(e, r) Hom( ,Ug(e, r)). U → ∗ Ug(e, r) is universal in the following sense. If Z is a scheme and φZ : g(e, r) U → Hom( ,Z) is a natural transformation, there exists a unique morphism γ : Ug(e, r) ∗ Z such that the transformations φZ and γ φU are equal. → ◦ Proof. Let e>e(g,r). Let µ : Sand on C satisfy (1) and (2) above. Note 10 C→ E µ (ω /S) is a locally free sheaf of rank N + 1 = 10(2g 2) g +1. Since s is ∗ C − − E nonspecial for each s S, µ ( ) is locally free of rank n = fe,r(0). Choose an open ∈ ∗ E 10 1 cover Wi of S trivializing both µ (ω )andµ ( ). Let Vi = µ− (Wi). For each { } ∗ /S ∗ E i, we obtain isomorphisms: C N+1 10 (31) C Wi =µ (ω /S) Wi ⊗O ∼ ∗ C | n (32) C Wi =µ ( )Wi. ⊗O ∼ ∗ E | These isomorphisms yield surjections: N+1 10 10 C Vi =µ∗µ (ω /S) Vi ω /S Vi 0, ⊗O ∼ ∗ C | → C | → n C Vi =µ∗µ ( )Vi Vi 0. ∼ ∗ ⊗O NE | →E| → The first surjection embeds Vi in Wi P . By the universal property of the Quot × scheme Qg(µ, n, fe,r) and the second surjection, there exists a map φi : Wi Qg(µ, n, fe,r). → r SS For t>t(g,r,e), φi(Wi) Q (µ, n, fe,r) . On the overlaps Wi Wj , φi and φj ⊂ g [k,1] ∩ differ by a morphism Wi Wj PSLN+1 PSLn ∩ → × corresponding to the choice of trivialization in (31) and (32). Hence there exists a well defined morphism φ : S Ug(e, r). → The functoriality of the universal property of the Quot scheme implies φ is func- torially associated to the data and µ : S. Wehaveshownthereexistsa natural transformation E C→ φU : g(e, r) Hom( ,Ug(e, r)). U → ∗ Suppose φZ : g(e, r) Hom( ,Z) is a natural transformation. There exists a U → ∗ r SS canonical element of δ g(e, r)(Q (µ, n, fe,r) ) corresponding to the universal ∈ U g [k,1] family on the Quot scheme. The morphism r SS φZ (δ):Q (µ, n, fe,r) Z g [k,1] → License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use A COMPACTIFICATION OVER Mg OF THE UNIVERSAL MODULI SPACE 459 is SLN+1 SLn-invariant. Hence φZ (δ) descends to γ : Ug(e, r) Z.Thetwo × → transformations φZ and γ φU agree on δ. Naturality now implies φZ = γ φU . ◦ ◦ By previous considerations, there are natural transformations tl : g(e, r) g(e + rl(2g 2),r) U → U − l given by ω /S. By Theorem (9.1.1), these induce natural isomorphisms E→E⊗ C tl : Ug(e, r) Ug(e + rl(2g 2),r). → − The arguments in the above proof imply a useful lemma: Lemma 9.1.1. Let µ : Sbe a flat family of Deligne-Mumford stable, genus C→ g 2 curves. Let be a µ-flat coherent sheaf on . The condition that s is a ≥ E C E slope-semistable torsion free sheaf of uniform rank on s is open on S. C Proof. Suppose s is a slope-semistable sheaf of uniform rank r on s for some E 0 C 0 s0 S. There exists an integer m such that ∈ (i) h1( ωm , ) = 0 for all s S. s s s (ii) E ω⊗m isC generatedC by global∈ section for all s S. s s (iii) degreeE ⊗ C( ωm ) >e(g,r). ∈ s0 s E ⊗ C 0 m It suffices to prove the lemma for ω /S.Letfe,r be the Hilbert polynomial m E⊗ C of ω /S. By the proof of Theorem (9.1.1), there exists an open set W S E⊗ C ⊂ containing s0 and a morphism φ : W Qg(µ, n = fe,r(0),fe,r) → m such that ω /S is isomorphic to the φ-pull back of the universal quotient. Since rE⊗ C SS φ(s0) Q (µ, n, fe,r) and the latter is open, the lemma is proven. ∈ g [k,1] 9.2. Deformations of torsion free sheaves and the irreducibility of Ug(e, r). We study deformation properties of uniform rank, torsion free sheaves on nodal curves. Lemma 9.2.1. Let µ : S Spec(C[t]) be a µ-flat, nonsingular surface. If is → E a µ-flat sheaf on S such that the restriction /t = 0 is torsion free, then 0 is locally free. E E E E 1 Proof. Let z µ− (0). Since S is a nonsingular surface, the local ring S,z is ∈ O regular of dimension 2. Consider the S,z-module z.Since is µ-flat, t is a z- O E E E regular element. Since 0 = /t is torsion free, depth S,z ( z) 2. We have the Auslander-Buchsbaum relationE E ([M]):E O E ≥ proj. dim S,z ( z)+depth S,z ( z )=dim( S,z)=2. O E O E O We conclude proj. dim S,z ( z)=0.Hence z is free over S,z. It follows that 0 is locally free. O E E O E Lemma (9.2.1) shows that it is not possible to deform a torsion free, non-locally free sheaf on a nodal curve to a locally free sheaf on a nonsingular curve if the de- formations at the nodes have local equations of the form (xy t) Spec(C[x, y, t]). However, the next lemma shows such deformations exist locally− if⊂ the deformations of the nodes have local equations of the form (xy t2). In Lemma (9.2.3), it is shown these local deformations can be globalized. − License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 460 RAHUL PANDHARIPANDE Lemma 9.2.2. Let S Spec(C[x, y, t]) be the subscheme defined by the ideal (xy t2).Letµ:S Spec⊂ (C[t]).Letζ=(0,0,0) S. There exists a µ-flat sheaf − → ∈ on S such that t=0 is locally free and 0 = mζ ,wheremζ is the maximal ideal E E 6 E ∼ defining ζ on S0. Proof. There exists a section L of µ defined by the ideal (x t, y t). Let be the coherent sheaf corresponding to this ideal. We have the exact− sequence:− E (33) 0 S L 0. →E→O →O → Since S is torsion free over C[t], so is . is therefore µ-flat. Since L is µ-flat,O sequence (33) remains exact afterE restrictionE to the special fiber. HenceO 0 = mζ . E ∼ Lemma 9.2.3. Let C be a Deligne-Mumford stable curve of genus g 2.LetE be a slope-semistable torsion free sheaf of uniform rank r on C. Then≥ there exists afamilyµ: 0 and a µ-flat coherent sheaf on such that : C→4 E C (1) 0 is a pointed curve. 4 (2) 0 = C, t =0 t is a complete, nonsingular, irreducible genus g curve. C ∼ ∀ 6 C (3) 0 = E, t =0 t is a slope-semistable torsion free sheaf of rank r. E ∼ ∀ 6 E Proof. Let z C be a node. Since E is torsion free and of uniform rank r, it follows from Propositions∈ (2) and (3) of chapter (8) of [Se]: az r (34) E = m , z ∼ Cz z 1 O ⊕ a +1 M Mz where mz is the localization of the ideal of the node z. To simplify the deformation arguments, let C be the field of complex numbers. Let B(d) C2 be the open ball ⊂ 2 of radius d with respect to the Euclidean norm; let B(d1,d2) C be the open ⊂ annulus. Disjoint open Euclidean neighborhoods, z Uz C, can be chosen for each node of C satisfying: ∈ ⊂ 2 (i) Uz is analytically isomorphic to B(dz) (xy =0) C . az r ∩ ⊂ (ii) E U = U mz. | z ∼ 1 O z ⊕ az +1 Let V U be the closed neighborhood of radius d /2. Let W = C V .A z z L L z z deformation⊂ of C can be given by the open cover: \∪ W 0 Defz z Cns , { ×4 }∪{ | ∈ } where Defz (to be defined below) is an open subset of 2 2 B(dz) 0 (xy t =0) C 0 ×4 ∩ − ⊂ ×4 containing (0, 0, 0). Defz is a local smoothing at the node z. Define dz dz 2 Kz = B , + z 0 (xy =0) C 0. 2 2 ×4 ∩ ⊂ ×4 Note B(r, s) is the annulus. Let µ denote the projection to 0.Forz>0(small 4 with respect to δz)and t <δz, it is not hard to find an isomorphism γz commuting with µ: || 2 γz : Kz Dz B(dz) 0 (xy t =0) → ⊂ ×4 ∩ − such that dz (35) B 0 Dz = 3 ×4 ∩ ∅ License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use A COMPACTIFICATION OVER Mg OF THE UNIVERSAL MODULI SPACE 461 and γz extends the identity map on the locus t = 0. Such a γz can be constructed by considering the holomorphic flow of an algebraic vector field on (xy t2 =0). The space − 2 B(dz) 0 (xy t =0) Dz ×4 ∩ − \ is disconnected. Let Defz be the complement of the component not containing (0, 0, 0). The isomorphism γz determines a patching of W 0 and Defz along ×4 Kz Dz in the obvious manner. The constructed family satisfies claims (1) and (2) of' the lemma. E0 can be extended trivially on W 0 to W . K is trivial by condition ×4 E| E| z (ii). By Lemma (9.2.2), mz can be flatly extended to a line bundle Lz on Defz. By (35) and the construction of Lemma (9.2.2), Lz canbeassumedtobetrivialon Dz.Bypatching az r Defz Lz 1 O ⊕ a +1 M Mz along Kz Dz, can be defined such that 0 ∼= E and t=0 is locally free. Indeed, such a patching' E exists for t = 0 by conditionE (ii). TheE patching6 can be extended trivially along Kz since Kz = Kz,t=0 0. ×4 Now condition (3) follows by Lemma (9.1.1). For a general ground field, the ´etale topology must be used. Proposition 9.2.1. Ug(e, r) is an irreducible variety. r SS Proof. Consider the morphism πSS : Q (µ, n = fe,r(0),fe,r) Hg.ByPropo- g [k,1] → sition (24) of chapter (1) of [Se], the scheme 1 SS πSS− ([C]) = Qg(C, n, fe,r ) 0 is irreducible for each nonsingular curve C,[C] Hg.SincethelocusHg Hg 1 0 ∈ ⊂ of nonsingular curves is irreducible, πSS− (Hg ) is irreducible. By Lemma (9.2.3), 1 0 r SS πSS− (Hg )isdenseinQg(µ, n, fe,r)[k,1]. Finally, since there is a surjective morphism r SS Q (µ, n, fe,r) Ug(e, r), g [k,1] → we conclude Ug(e, r) is irreducible. 9.3. The normality of Ug(e, 1). We need an infinitesimal analogue of Lemma (9.2.1). First, we establish some notation. Let R = C[[x, y]][]/(xy , 2), − Let A = R/R ∼= C[[x, y]]/(xy). Let m =(x, y) be the maximal ideal of A.There is a natural, flat inclusion of rings C[]/(2) R.ForaC[]/(2)-module M,let denote the M endomorphism given by multiplication→ by . ∗ Lemma 9.3.1. There does not exist a C[]/(2)-flat R-module E such that E/E ∼= m as A-modules. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 462 RAHUL PANDHARIPANDE Proof. Suppose such an E exists. Let α : E E/E ∼ m. → → Let ex, ey E satisfy α(ex)=xand α(ey)=y. We obtain a morphism ∈ β : R R E ⊕ → defined by β(1, 0) = ex and β(0, 1) = ey.Sinceis nilpotent, β is surjective by Nakayama’s Lemma. We have an exact sequence 0 N R R E 0. → → ⊕ → → Since E and R R are C[]/(2)-flat, N is C[]/(2)-flat. Hence, there is an exact sequence ⊕ (36) 0 N N ∗ N → → → obtained by tensoring 0 () C[]/(2) ∗ C[]/(2)withN.Sinceβ(y,0) E, there exists an element→ n =(→y+f(x, y→),g(x, y)) in N.Considerxn N: ∈ ∈ xn =(xy + xf(x, y),xg(x, y)) = ((1 + xf(x, y)),xg(x, y)). Note xn = 0. By the exactness of (36), there exists an n N satisfying n = xn. Since R R is flat over C[]/(2), any such n must be of the∈ form ⊕ n =(1+xf(x, y)+fˆ(x, y),xg(x, y)+gˆ(x, y)). Since α β(n)=x+x2f(x, y) =0inm, ncannot lie in N. We have a contradiction. No such◦E can exist. 6 1 SS To prove Ug (e, 1) is normal, it suffices to show Qg(µ, n = fe,1,fe,1)[k,1] is nonsingular. The nonsingularity is established by computing the dimension of 1 SS Qg(µ, n, fe,1)[k,1] and then bounding the dimension of the Zariski tangent space at each point. The Zariski tangent spaces are controlled by a study of the differen- tial dπSS,whereπSS is the canonical map 1 SS πSS : Q (µ, n, fe,1) Hg. g [k,1] → 1 SS Lemma 9.3.2. Qg(µ, n = fe,1(0),fe,1)[k,1] is nonsingular. 1 SS Proof. Consider the universal quotient sequence over Qg(µ, n, fe,1)[k,1], n 0 C Qss U 0. →F→ ⊗O × →E→ 1 SS Let ξ Q (µ, n, fe,1) be a closed point and let C = U . ξ corresponds to a ∈ g [k,1] π(ξ) quotient n (37) 0 ξ C C ξ 0. →F → ⊗O →E → 1 There is a natural identification of the Zariski tangent space to Qg(C, n, fe,1)atξ: 1 0 Tξ(Q (C, n, fe,1)) = H (C, Hom( ξ, ξ)) g ∼ F E 1 (see [Gr]). If h (C, Hom( ξ, ξ)) = 0 and the deformations of F E n (38) C C ξ 0 ⊗O →E → 1 are locally unobstructed, then ξ is a nonsingular point of Qg(C, n, fe,1). If ξ is locally free, the deformations of (38) are locally unobstructed. Sequence (37) yields:E (39) n 1 0 Hom( ξ, ξ) Hom(C C, ξ) Hom( ξ, ξ) Ext ( ξ, ξ) 0. → E E → ⊗O E → F E → E E → License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use A COMPACTIFICATION OVER Mg OF THE UNIVERSAL MODULI SPACE 463 1 Since Ext ( ξ, ξ) is torsion and E E 1 n 1 h (C, Hom(C C, ξ)) = n h (C, ξ)=0, ⊗O E · E 1 we obtain h (C, Hom( ξ, ξ)) = 0. F E From (34), at each node z C, ξ is either locally free or locally isomorphic ∈ E to mz.Letsbe the number of nodes where ξ,z = mz. Using (39), we compute E ∼ χ(Hom( ξ, ξ)) in terms of s: F E n 1 χ(Hom( ξ, ξ)) = χ(Hom( ξ, ξ)) + χ(Hom(C C, ξ)) + χ(Ext ( ξ, ξ)). F E − E E ⊗O E E E n 2 It is clear χ(Hom(C C, ξ)) = n . Let A = C[[x, y]]/(xy)andm=(x, y) A as above. It is not hard⊗O to establish:E ⊂ 1 2 (40) ExtA(m, m)=C , i (41) 0 A HomA(m, m) C 0 → → → → 1 where i is the natural inclusion. Since Ext ( ξ, ξ) is a torsion sheaf supported at E E each z C where ξ,z = mz with stalk (40), ∈ E ∼ 1 χ(Ext ( ξ, ξ)) = 2s. E E There is a natural inclusion i 0 C Hom( ξ, ξ) δ 0. →O → E E → → Since ξ is of rank 1, δ is a torsion sheaf supported at the nodes where ξ is not locallyE free. At these nodes, δ can be determined locally by (41). Hence E χ(Hom( ξ, ξ)) = 1 g + s. E E − Summing the Euler characteristics yields: 0 2 2 h (C, Hom( ξ, ξ)) = g 1 s + n +2s=n 1+g+s. F E − − − If C is a nonsingular curve, ξ is locally free on C. The above results show that E1 1 SS 2 ξ is a nonsingular point of Q (C, n, fe,1). Thus dim(Q (µ, n, fe,1) )=n 1+ g g [k,1] − g+dim(Hg). 1 SS Let ξ Q (µ, n, fe,1) be a closed point with the notation given above. We ∈ g [k,1] examine the exact differential sequence: 1 1 SS dπSS 0 Tξ(π− [C]) Tξ(Q (µ, n, fe,1) ) T (Hg). → SS → g [k,1] → [C] Recall Hg is nonsingular. By Lemma (9.3.1) and the surjection of T[C](Hg)onto the miniversal deformation space of C, dim(im(dπSS)) dim(Hg) s ≤ − (s is the number of nodes where ξ is not locally free). By previous results: E 1 2 dim(Tξ(π− [C])) = n 1+g+s. SS − It follows: 1 SS 2 (42) dim(Tξ(Q (µ, n, fe,1) )) (n 1+g+s)+(dim(Hg) s) g [k,1] ≤ − − 1 SS = dim(Qg(µ, n, fe,1)[k,1]). Equality must hold in equation (42). ξ is therefore a nonsingular point of 1 SS Qg(µ, n, fe,1)[k,1]. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 464 RAHUL PANDHARIPANDE As a consequence, we obtain: Proposition 9.3.1. Ug(e, 1) is normal. 10. The isomorphism between Ug(e, 1) and Pg,e 10.1. AreviewofPg,e. In this section, the compactification of the universal Pi- card variety of degree e line bundles, Pg,e, described in [Ca] is considered. Let e be large enough to guarantee the existence and properties of Pg,e and Ug(e, 1). A natural isomorphism ν : Pg,e Ug(e, 1) will be constructed. We follow the notation of→ [Ca], [Gi]. A Deligne-Mumford quasi-stable,genusg curve C is a Deligne-Mumford semistable, genus g curve with destabilizing chains of length at most one. Let ψ : C Cs be the canonical contraction to the Deligne- Mumford stable model. For each→ complete subcurve D C,letDc =C D. Define: ⊂ \ c kD =#(D D ). ∩ Let ωC,D be the degree of the canonical bundle ωC restricted to D.LetLbe a degree e line bundle on C. Denote by LD the restriction of L to D.LeteDbe the degree of LD. L has (semi)stable multidegree if for each complete, proper subcurve D C, the following holds: ⊂ ωC,D (43) eD e ( ) Zg,e.LetL= UZ(1) and = ψ (L). In Lemma (10.2.6), is shown to be a flat family of slope-semistableO E torsion∗ free sheaves of uniform rankE 1 and degree e over Zg,e. Care is required in establishing flatness. The argument depends upon Zariski’s theorem on formal functions and the criterion of Lemma (10.2.5). By the License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use A COMPACTIFICATION OVER Mg OF THE UNIVERSAL MODULI SPACE 465 universal property of Ug(e, 1), induces a map νZ : Zg,e Ug(e, 1). Since νZ is E → SLM+1-invariant, a map ν : Pg,e Ug(e, 1) is obtained. → It remains to prove ν is an isomorphism. Since Ug(e, 1) is normal by Proposi- tion (9.3.1), it suffices to show ν is bijective. Surjectivity is clear. Injectivity is established in section (10.3). 10.2. The construction of ν. Multidegree (semi)stability corresponds to slope- (semi)stability in the following manner: Lemma 10.2.1. Let C be a Deligne-Mumford quasi-stable curve. If L is a very ample, degree e line bundle on C of (semi)stable multidegree, then E = ψ (L) is a ∗ slope-(semi)stable, torsion free sheaf of uniform rank 1 and degree e on Cs.Also, if L is of strictly semistable multidegree, then E is strictly slope-semistable. First we need a simple technical result. For each complete subcurve D of C, define the sheaf FD on C by the sequence: 0 FD L LDc 0. → → → → FD is the subsheaf of sections of L with support on D.Infact,FDis exactly the c subsheaf of sections of LD vanishing on D D . Therefore degree(FD)=eD kD. ∩ − Note by Riemann-Roch, χ(FD)=degree(FD)+1 gD,wheregD is the arithmetic genus of D. We obtain, − (44) eD = χ(FD)+gD 1+kD. − Lemma 10.2.2. Let C be a Deligne-Mumford quasi-stable curve. Let L be a very ample line bundle of semistable multidegree. For every complete subcurve D C, 1 ⊂ R ψ (FD)=0. ∗ Proof. Afiberofψis either a point or a destabilizing P1. Byinequality(43) 1 and ampleness, the restriction of L to a destabilizing P is P1 (1). Let P be a destabilizing P1 of C. There are five cases: O c (i) P D , P D = .ThenFDP =0. ⊂ c ∩ ∅ | (ii) P D , P D = .ThenFDP is torsion. ⊂ ∩ c 6 ∅ | (iii) P D, P D = .ThenFDP = P(1). ⊂ ∩ c∅ | O (iv) P D,#P D =1.ThenFD P = P. ⊂ | ∩ c| | O (v) P D,#P D =2.ThenFD P = P( 1). ⊂ 1| ∩ | | O −1 In each case, h (P, FD P ) = 0. The vanishing of h (P, FD P ) and the simple char- 1 | | acter of ψ imply R ψ (FD)=0. ∗ Proof. [Of Lemma (10.2.1)] Let C1 be the union of the destabilizing P1’s of C.By 1 1 Lemma (10.2.2), R φ (FC1 ) = 0. Since each destabilizing P in C is of type (v) in ∗ the proof of Lemma (10.2.2), FC1 P = P ( 1). It follows that ψ (FC1 )=0.By the long exact sequence associated| to O − ∗ 0 F 1 L L 1c 0, → C → → C → 1c it follows E = ψ (LC1c ). Since LC1c is torsion free on C and the morphism 1c ∼ ∗ C Cs is finite, E is torsion free. Certainly, E is of uniform rank 1. → Let 0 G E be a proper subsheaf. Let Ds Cs be the support of G. → → ⊂ If Ds = Cs, the inequality of slope-stability follows trivially. We can assume Ds 1 is a complete, proper subcurve. Let D = ψ− (Ds). D C is a complete, proper ⊂ License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 466 RAHUL PANDHARIPANDE subcurve. It is clear that G is a subsheaf of ψ (FD) with torsion quotient. Therefore, ∗ it suffices to check slope-(semi)stability for ψ (FD). Certainly ∗ 0 0 h (Cs,ψ (FD)) = h (C, FD). ∗ 1 By Lemma (10.2.2), R ψ (FD) = 0. Thus by a degenerate Leray spectral sequence ([H], Ex. 8.1, p.252), ∗ 1 1 h (Cs,ψ (FD)) = h (C, FD). ∗ Hence χ(FD)=χ(ψ(FD)). Similarly χ(L)=χ(ψ(L)). Inequality (43) and equation (44) yield: ∗ ∗ 2gD 2+kD (χ(FD)+gD 1+kD) (χ(L)+g 1) − ( ) χ(ψ (FD)) χ(ψ (L)) ∗ ( ) < ∗ . ωD ,C ≤ 2g 2 s s − Hence, ψ (L) is slope-(semi)stable. The final claim about strict semistability also follows from∗ the proof. s Let ψ : UZ UZ , L = UZ (1), and = ψ (L) be as defined in section (10.1). → O E ∗ 1 We now establish that is flat over Zg,e. The vanishing of R ψ (L)isprovedby Zariski’s theorem on formalE functions in Lemma (10.2.4). The flatness∗ criterion of Lemma (10.2.5) is then applied to obtain the required flatness. First we need an auxiliary result. Let [C] Ze,g and let P UZ be a destabi- 1 ∈ ⊂ lizing P of C. The conormal bundle, NP∗ ,ofPin UZ is locally free (P , UZ are 1 nonsingular). Recall a locally free sheaf 1 (ai)onP is said to be non-negative OP if each ai 0. ≥ L Lemma 10.2.3. NP∗ is non-negative. Proof. Let TU and TZ denote the tangent sheaves of UZ and Ze,g.Letρ:UZ Ze,g be the natural morphism. There is a differential map → dρ : TU ρ∗(TZ ). → Restriction to P yields a (non-exact) sequence 0 TP TU P ρ∗(TZ ) P . → → | → | Certainly, ρ∗(TZ ) P = P .Weobtainamap | ∼ O L α:NP P. → O Let Pˆ P be the locus of nonsingularM points of C. Since the morphism ρ is ⊂ smooth on Pˆ, α ˆ is an isomorphism of sheaves. Since NP is a torsion free sheaf, |P α must be an injection of sheaves. It follows easily NP is non-positive. Hence NP∗ is non-negative. In fact, an examination of the deformation theory yields NP∗ ∼= P (1) P (1) I where I is a trivial bundle. We will need only the non-negativityO result.⊕O ⊕ Lemma 10.2.4. R1ψ (L)=0. ∗ License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use A COMPACTIFICATION OVER Mg OF THE UNIVERSAL MODULI SPACE 467 Proof. Let ζ U s . It suffices to prove ∈ Z 1 R ψ (L)ζ =0 ∗ in case ζ is a node of stable curve destabilized in UZ .Letmbe the ideal of ζ. 1 1 ψ− (m) is the ideal of the nonsingular destabilizing P = P .LetPndenote the 1 n 1 n subscheme of UZ defined by ψ− (m ) ∼= ψ− (m) .LetLnbe the restriction of L to Pn. By Zariski’s Theorem on formal functions: 1 lim 1 R ψ (L)ζˆ = H (Pn,Ln). ∗ ∼ ← Since completion is faithfully flat for noetherian local rings, it suffices to show for 1 each n 1, h (Pn,Ln) = 0. As above, denote the conormal bundle of P in UZ by ≥ NP∗ . Since the varieties in question are nonsingular, there is an isomorphism on P : n 1 n n 1 m − /m ∼= Sym − (NP∗ ). 1 Since the pair (C, LC ) is of semistable multidegree and P is a destabilizing P , 1 L1 ∼= P (1). Hence h (P1,L1)=0.There is an exact sequence on Pn for each n 2:O ≥ n 1 n 0 m − /m Ln Ln Ln 1 0. → ⊗ → → − → There is a natural identification n 1 n n 1 m − /m Ln = Sym − (NP∗ ) P P (1). ⊗ ∼ ⊗O O From the non-negativity of NP∗ (Lemma (10.2.3)), we see 1 n 1 n h (Pn,m − /m Ln)=0. ⊗ By the induction hypothesis 1 1 h (Pn,Ln 1)=h (Pn 1,Ln 1)=0. − − − 1 Thus h (Pn,Ln) = 0. The proof is complete. Lemma 10.2.5. Let φ : B1 B2 be a projective morphism of schemes over A.If → i Fis a sheaf on B1 flat over A and i 1, R φ (F )=0,thenφ(F)is flat over A. ∀ ≥ ∗ ∗ k Proof. We can assume A and B2 are affine and B1 = P .Let be the standard ∼ B2 U k + 1 affine cover of B2. There is a Cech resolution computing the cohomology of F on B1: 0 0 1 k (45) 0 H (B1,F) C ( ,F) C ( ,F) ... C ( ,F) 0. → → U → U → → U → Since Riφ (F )=0fori 1, the resolution (45) is exact. Since F is A-flat, the Cech ∗ j ≥ 0 modules C ( ,F)areallA-flat. Exactness of (45) implies H (B1,F) ∼= φ (F)is A-flat. U ∗ Lemma 10.2.6. is a flat family of slope-semistable, torsion free sheaves of uni- E form rank 1 over Ze,g. Proof. By Lemmas (10.2.4) and (10.2.5), is flat over Ze,g. Let [C] Ze,g.We have a diagram: E ∈ iC C U Z −−−−→ (46) ψ C ψ s Cs UZ y −−−−i C s → y License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 468 RAHUL PANDHARIPANDE If i∗ ( ) = ψC (i∗ (L)), then the proof is complete by Lemma (10.2.1). There is a Cs ∼ C naturalE morphism∗ of sheaves γC : iC∗ ( ) ψC (iC∗ (L)). s E → ∗ We first show γC is a surjection. Since, by Lemma (10.2.1), ψC (iC∗ (L)) is a slope- ∗ semistable torsion free sheaf of degree e, ψC (iC∗ (L)) is generated by global sections. There is a natural identification ∗ 0 0 H (Cs,ψC (iC∗ (L))) = H (C, C (1)). ∗ ∼ O 0 By the nondegeneracy of C and the nonspeciality of C (1), H (C, C (1)) is canon- 0 M O O ically isomorphic to H (P , PM (1)). These sections extend over UZ and thus appear in i ( ). Therefore, γO is a surjection. C∗ s C E 1 Since ψ is an isomorphism except at destabilizing P ’s of UZ,thekernelofγC is a torsion sheaf on Cs. Flatness of over Ze,g implies χ(i∗ ( )) is independent of E Cs E [C] Ze,g. By Lemma (10.2.1), χ(ψC (iC∗ (L))) is independent of [C] Ze,g.Over ∈ ∗ ∈ the open locus of nonsingular curves [C] Ze,g, ψ is an isomorphism thus: ∈ (47) χ(iC∗ ( )) = χ(ψC (iC∗ (L))). s E ∗ By the above considerations, (47) holds for every [C] Ze,g. Hence, the torsion ∈ kernel of γC must be zero. γC is an isomorphism. The proof is complete. By combining Lemma (10.2.6) with Theorem (9.1.1), there exists a natural morphism νZ : Ze,g Ug(e, 1). Since νZ is SLM+1-invariant, νZ descends to → ν : Pg,e Ug(e, 1). Certainly ν is surjective. Since Ug(e, 1) is normal, ν is an isomorphism→ if and only if ν is injective. The injectivity of ν will be established in section (10.3). 10.3. Injectivity of ν. Let C be a Deligne-Mumford quasi-stable genus g curve. Let D C be a complete subcurve. A node z D is an external node of D if z Dc⊂. P D is a destabilizing P1 of D if P∈ is a destabilizing P1 of C.A destabilizing∈ ⊂P1 of D is external if P Dc = .LetDˆdenote D minus all the external destabilizing P1’s of D. ∩ 6 ∅ (C, L)isasemistable pair if C is Deligne-Mumford quasi-stable and L is a very ample line bundle of semistable multirank. The semistable pairs (C, L)and(C0,L0) are isomorphic if there exists an isomorphism γ : C C0 such that γ∗(L0) ∼= L.A complete subcurve D C is an extremal subcurve of→ the semistable pair (C, L)if equality holds in (43):⊂ ωC,D eD e = kD/2. − · 2g 2 − The semistable pair (C, L)issaidtobemaximal if the following condition is sat- isfied: if z C is an external node of an extremal subcurve, z is contained in a ∈ 1 destabilizing P .Letψ:C Csbe the stable contraction. By Lemma (10.2.1), ψ (L) is a slope-semistable, torsion→ free sheaf of uniform rank 1. Let J(ψ (L)) be the∗ associated set of slope-stable Jordan-Holder factors. ∗ (C, J)isaJordan-Holder pair if C is a Deligne-Mumford stable curve and J is a set of slope-stable, torsion free sheaves. As before, the Jordan-Holder pairs (C, J) and (C0,J0) are isomorphic if there exists an isomorphism γ : C C0 such that → γ∗(J 0) ∼= J. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use A COMPACTIFICATION OVER Mg OF THE UNIVERSAL MODULI SPACE 469 Lemma 10.3.1. Let (C, L), (C0,L0)be maximal semistable pairs. If (Cs,J(ψ (L))) ∗ and (Cs0 ,J(ψ0(L0))) are isomorphic Jordan-Holder pairs, then (C, L) and (C0,L0) are isomorphic∗ semistable pairs. Proof. Consider a Jordan-Holder filtration of E = ψ (L)onCs: ∗ 0=E0 E1 E2 ... En =E. ⊂ ⊂ ⊂ ⊂ Let Ai = Supp(Ei). For each 1 i n, ≤ ≤ χ(E ) χ(E) i = . wA ,C 2g 2 i s − 1 By the proof of Lemma (10.2.1), we see the Bi = ψ− (Ai) are extremal subcurves of (C, L)for1 i n. As before, let FB be the subsheaf of sections of L with ≤ ≤ i support on Bi. From the proof of Lemma (10.2.1), it follows Ei = ψ (FBi ). For ∼ ∗ 1 i n,letXi =Ai Ai 1 and Yi = Bi Bi 1.WeseeSupp(Ei/Ei 1)=Xi. ≤ ≤ \ − \ − − If z Xi is an internal node of Xi, z is destabilized by ψ if and only if (Ei/Ei 1) ∈ − is locally isomorphic to mz at z.Ifz Xiis an external node, then there are two 1 ∈1 c 1 1 cases. If z Ai 1,thenψ− (z)=P Yi. If z Ai ,thenψ− (z)=P Yi. ∈ − ∼ 6⊂ ∈ ∼ ⊂ These conclusions follow from the maximality of (C, L). It is now clear Cs and the Jordan-Holder factor Ei/Ei 1 determine Yˆi completely. Also, the Yˆi are connected 1 − by destabilizing P ’s. We have shown (Cs,J) determines C up to isomorphism. We will show below in Lemmas (10.3.2-10.3.3) that L is determined up to isomorphism Yˆi 1 by Ei/Ei 1. Since the Yˆi are connected by destabilizing P ’s, the isomorphism class of the pair− (C, L) is determined by the line bundles L . This completes the proof Yˆi of the lemma. Lemma 10.3.2. There is an isomorphism ψ (L ˆ ) = Ei/Ei 1. ∗ Yi ∼ − Proof. We keep the notation of the previous Lemma. Certainly Supp(FBi/FBi 1 )= c − Yi.Letpibe the divisor Bi 1 Yi Yi.Letqibe the divisor Bi Yi Yi.Since − ∩ ⊂1 ∩ ⊂ the points of pi lie on destabilizing P ’s joining Yi to Yi 1 and the points of qi lie on 1 − destabilizing P ’s joining Yi to Yi+1,wenotepi qi = . There is an isomorphism ∩ ∅ LY = FY Y (pi +qi). Since there is an exact sequence: i ∼ i ⊗O i 0 FYi (FBi /FBi 1 ) pi 0, → → − → → we see FYi =(FBi/FBi 1 ) Yi( pi). − ⊗O − Thus LYi = (FBi /FBi 1 ) Yi(qi). ∼ − ⊗O Since Bi is an extremal subcurve of C and the pair (C, L) is maximal, we see qi 1 lies on external destabilizing P ’s of Yi. Hence L ˆ is isomorphic to (FBi /FBi 1 ) ˆ . Yi − Yi We have an exact sequence on Yi: 0 IYˆ (FBi /FBi 1 ) (FBi /FBi 1 ) (FBi /FBi 1 )Yˆ 0. → i ⊗ − → − → − i → 1 Let P be an external destabilizing P of Yi.IfPmeets Bi 1 then P Bi 1.Thus c − ⊂ − each such P meets Bi . ItisnownothardtoseeIYˆ (FBi/FBi 1 ) restricts to i ⊗ − P ( 1) on each such P . Therefore, by familiar arguments, O − ψ ((FBi /FBi 1 )Yˆ ) = ψ (FBi /FBi 1 ). ∗ − i ∼ ∗ − License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 470 RAHUL PANDHARIPANDE 1 By Lemma (10.2.2), R ψ (FBi 1 )=0.Hence ∗ − ψ (FBi/FBi 1 ) = (ψ (FBi )/ψ (FBi 1 )) = Ei/Ei 1. ∗ − ∼ ∗ ∗ − ∼ − Following all the isomorphisms yields the lemma. Lemma 10.3.3. Let (C, L) be a semistable pair. Let D C satisfying Dˆ = D. ⊂ Then LD is determined up to isomorphism by ψ (LD). ∗ Proof. Let D1 be the union of the destabilizing P1’s of D. Note all these P1’s are 1 internal. Let D0 = D D and denote the restriction of ψ to D0 by ψ0.Consider the sequence on D: \ 0 ID0 LD LD LD0 0. → ⊗ → → → 1 Since ID LD restricts to 1 ( 1) on each destabilizing P of D,wesee 0 ⊗ OP − 1 ψ (ID LD)=R ψ (ID LD)=0. ∗ 0 ⊗ ∗ 0 ⊗ Thus ψ0 (LD0 ) = ψ (LD0 ) = ψ (LD). Since ψ0 is a finite affine morphism, ∗ ∼ ∗ ∼ ∗ β : ψ0∗(ψ0(LD0)) LD0 0. ∗ → → Let τ be the torsion subsheaf of ψ0∗(ψ0(LD0)). Since β is generically an isomor- ∗ phism and LD0 is torsion free on D0,wesee LD0 =(ψ0∗(ψ0(LD0))/τ). ∼ ∗ We have shown that LD is determined up to isomorphism by ψ (LD). It is clear 0 ∗ that LD0 determines LD up to isomorphism. Let ρ : Ze,g Pg,e be the quotient map. Let ζ Pg,e. It follows from the results → ∈ of [Ca] (Lemma 6.1, p. 640) that there exists a [C] Zg,e satisfying: ∈ (i) ρ([C]) = ζ. (ii) (C, L = C (1)) is a maximal semistable pair. O Let ψC : C Cs be the stable contraction. Let E = ψ (L). Let J be the Jordan- → ∗ Holder factors of E on Cs. From the definition of ν, ν(ζ) is the element of Ug(e, 1) corresponding to the isomorphism class of the data (Cs,J). By Lemmas (10.3.2- 10.3.3), the isomorphism class of (C, L) is determined by the isomorphism class of (Cs,J). Therefore ν is injective. By the previous discussion, ν is an isomorphism. 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Newstead, Introduction to Moduli Problems and Orbit Spaces, Tata Institute Lecture Notes, Springer-Verlag, 1978. MR 81k:14002 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use A COMPACTIFICATION OVER Mg OF THE UNIVERSAL MODULI SPACE 471 [Se] C. Seshadri, Fibr´es Vectoriels sur les Courbes Alg´ebriques,Ast´erisques 96,Soci´et´e Math´ematique de France, 1982. MR 85b:14023 [Si] C. Simpson, Representations of the Fundamental Group of a Smooth Projective Variety, preprint. [T] M. Teixidor, in preparation. Department of Mathematics, University of Chicago, 5734 S. University Ave., Chicago, Illinois 60637 E-mail address: [email protected] License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use